Nine days in Tokyo and Seoul: two capitals that stay open later than you can

A Shinto shrine to the Meiji emperor, finished in 1920 and rebuilt after the war, standing in seventy hectares of forest that was planted from scratch with a hundred thousand trees donated from all over Japan. The approach under the enormous cypress torii takes ten minutes and the traffic disappears entirely. Free, and open from dawn to dusk.

Three hundred metres of pedestrian lane straight out of Harajuku station and the exact opposite of what you have just walked through: crepe stands, cosmetics, photo booths and a great many fourteen year olds. It takes fifteen minutes and it is one of the more honest things in Tokyo.

The scramble outside Shibuya station lets pedestrians cross in every direction at once and moves something like two thousand people on a busy signal change. Cross it, then watch it from the walkway above the station or from a window seat in one of the cafes on the north west corner. Better after dark.

The alleys behind both stations are lined with six seat counters doing one thing well: yakitori over charcoal, tonkatsu, standing sushi. Look for a shop with a queue and a plastic model in the window, and pay at a machine by the door if there is one.

The Thunder Gate, with a four metre paper lantern hung in the middle of it and a wind god and a thunder god in the alcoves either side. It has burned down repeatedly since the tenth century and the present one is a 1960 concrete rebuild paid for by the founder of Panasonic, whose name is on the bottom of the lantern.

Two hundred and fifty metres of covered stalls running from the gate up to the temple, selling rice crackers, fans, and sweet bean cakes cooked in front of you. Traders have had licences here for three hundred years. At eight in the morning half the shutters are still down and you can see the painted panels on them, which are worth more than most of the stock.

This is a tempura and soba neighbourhood and several shops have been at it for over a century. Otherwise monjayaki, a runnier Tokyo cousin of the Osaka pancake, cooked on the table and scraped up with a small spatula.

Six hundred and thirty four metres of lattice steel finished in 2012, built as a broadcast tower because the old one was too short for the buildings that had grown around it. Two decks, at 350 and 450 metres, with a glass floor at the lower one. Book online for a timed slot; the queue on the day is long and the last hour of light is what you want.

The inner citadel of Edo Castle, open free most days and almost empty compared with everything around it. The stone base of the keep is still there, thirty metres of dressed granite with nothing on top: it burned in 1657 and the shogunate decided the money was better spent rebuilding the city. There is a formal Japanese garden below it and a bamboo grove of species from all over the country.

The basement levels of the station and the buildings around it are full of proper restaurants rather than fast food, and the lunch sets between eleven and two are the cheapest good eating in central Tokyo.

The oldest and largest museum in Japan, six buildings around a courtyard in Ueno park. The main hall runs chronologically from prehistory to the nineteenth century, and the smaller gallery of national treasures rotates every few weeks because the scrolls and screens cannot be lit for long. Two hours is the minimum; the swords and the Jomon pottery are the two rooms people remember.

Across the park, and worth twenty minutes even if Western art is the last thing you came to Japan for: the building is by Le Corbusier, one of seventeen of his works listed together by UNESCO, and it was built to house a collection a Japanese industrialist bought in Paris before the war and the French government held onto until 1959.

Eleven and a half metres of bronze cast in 1252, sitting outdoors because the hall built over it was destroyed by storms twice and finally taken away by a tsunami in 1498, after which nobody rebuilt it. It is hollow and for a small extra fee you can go inside and see the casting seams. The face is the thing: it was designed to be looked up at from close range and it works.

Ten minutes downhill, a temple built into a wooded slope around a nine metre gilded wooden Kannon with eleven heads. The terrace halfway up looks out over the town to the sea, there is a cave at the bottom with carvings cut into the walls, and in June the hydrangea path up the hillside is timed entry because of the numbers.

Shirasu, tiny whitebait, is the local dish, served raw or boiled over rice, and every second restaurant along the coast road does a bowl of it. The beach itself is grey sand and full of surfers.

The shrine at the head of the town's main avenue, founded in 1063 and moved here in 1180 by the first shogun, with a wide stone approach running a kilometre and a half down to the sea. Climb the steps to the upper hall for the view back along that axis. A great many weddings happen here on Sundays.

A little over two hours in the air. Gimpo is on the metro and the airport railway, twenty to thirty minutes from the centre. Have the K-ETA sorted before you fly if your passport needs one.

The one palace with Western buildings in it: a stone hall with columns, built in 1910 when the Korean Empire was trying very hard to be taken seriously, standing beside traditional wooden halls in the same courtyard. The emperor Gojong took refuge here after his wife was murdered by Japanese agents in 1895. It opens late, and it is floodlit in the evening.

An elevated motorway approach over Seoul Station, closed to traffic and replanted as a walkway in 2017, with several hundred species of tree and shrub in concrete tubs along a kilometre of it. It joins two halves of the city that the road had separated for forty years, and it is best at dusk.

The old printing and lighting district east of the palace has filled up with grill houses in converted workshops. Order samgyeopsal, pork belly cooked on a burner at the table, and the free side dishes will keep arriving whether you eat them or not.

Built in 1395 as the main palace of the Joseon dynasty, burned in the Japanese invasion of 1592, left as ruins for nearly three centuries, rebuilt in 1867 and then mostly pulled down again under Japanese colonial rule, which put its own government building across the front of it. The restoration has been running since 1990 and is not finished. The throne hall and the banqueting pavilion on the lotus pond are the two things to slow down for.

Hundreds of hanok, tiled wooden courtyard houses around small courtyards, on the slope between the two palaces. Most were built in the 1920s and 1930s as speculative housing and nearly all of them are lived in now. Signs ask visitors to keep quiet and some lanes close to sightseers in the evening, which is a fair request from people whose front doors you are photographing.

The street along the palace wall is lined with small restaurants in converted houses. Kalguksu, hand cut wheat noodles in a broth, or a bibimbap set that arrives with a dozen side dishes and no warning.

Begun in 1405 as a secondary palace and preferred by the kings to the main one for most of three centuries, because it follows the contours of the hill instead of forcing a grid onto them. Behind it is the Huwon, a wooded garden of ponds and pavilions on the World Heritage list, entered only on a timed guided tour and worth arranging your day around.

A low hill on the eastern ridge with the restored city wall running over the top of it and a path along the inside face. The village on the slope below was a poor quarter painted with murals in the 2000s, which brought the tourists and a good deal of resentment with them. From the top you get the whole basin, the palaces on one side and the towers on the other.

Zaha Hadid's aluminium building of 2014, a single continuous curved surface with no straight lines and no obvious front, set over an excavated run of the old fortification and a Joseon military camp that were found during construction and left on show. Exhibition halls inside, a park on the roof, and an LED rose field lit at night.

One of the oldest permanent markets in Korea, opened in 1905, and the food alley through the middle of it is the reason to come. Bindaetteok, mung bean pancakes ground and fried in front of you, mayak gimbap, and raw beef tartare from the stalls at the far end. Sit at a stool at the counter and point. Busiest and best from about six.

Two collections in one complex in Itaewon, built by three different architects: traditional Korean art, including celadon and Joseon painting, in a terracotta block by Mario Botta, and contemporary work in the buildings by Jean Nouvel and Rem Koolhaas next to it. The old Korean half is the better one and the audio guide is unusually good.

Itaewon is the one part of Seoul with a serious range of foreign food, which after four days of Korean cooking is either a relief or a waste. If you would rather stay local, naengmyeon, cold buckwheat noodles in iced broth, is the summer dish and a northern one.

A working Buddhist temple founded in 794 and rebuilt many times, now sitting directly opposite the convention centre with glass towers on three sides. Monks chant in the main hall while the traffic goes past. There is a twenty three metre stone Maitreya at the top of the steps and a temple stay programme for anybody who wants the full four in the morning version.

Five hundred and fifty five metres and a hundred and twenty three floors, finished in 2017 and the tallest building in Korea by a long way. The observatory occupies the top few floors with a glass floored section, and the lift is one of the fastest in the world. Book a timed slot and go for the last hour of daylight.

Korea's first Gothic brick cathedral, finished in 1898 on a site Catholic converts had used secretly for a century when the religion was banned and its followers executed. In the 1970s and 1980s it became the place dissidents sheltered from the military government, which is why it means more here than a nineteenth century church usually would.

Kalguksu and dumplings from one of the old shops in the back lanes, or a bowl of seolleongtang, milky ox bone soup that is served with rice and a bowl of kimchi and is what Seoul eats when it is cold or hungover.