Three days in Alexandria: an ancient city you visit by going down

Trains from Cairo come into Misr Station, which is a ten minute walk from the first site of the trip, so this day works even if you leave Cairo late morning.

Kom el-Dikka is an entire Roman quarter in the middle of the city, uncovered in 1960 when the rubble mound that gave the place its name was cleared for building work, and dug by a Polish mission ever since. There is a small marble theatre with thirteen tiers of seating, a bath complex, lecture halls belonging to what was probably a late antique university, and the Villa of the Birds, whose second century floor mosaics of peacocks, pigeons and a panther are kept under a shelter and are the best thing on the site.

In the afternoon, out along the eastern harbour to the Bibliotheca Alexandrina, which opened in 2002 on ground near where the ancient library is thought to have stood. The building is the reason to go: a huge tilted granite disc, its outer wall carved with characters from scripts all over the world, and inside a single stepped reading room descending in eleven levels. There are museums of antiquities and manuscripts inside, and a planetarium in the sphere outside.

Then walk back along the corniche as the light drops, and eat fish.

Two sites in the old south-western quarter first, five minutes apart. The catacombs of Kom el Shoqafa are cut three levels into the rock and were reached by a spiral staircase around a shaft, down which the bodies were lowered. They were begun as one wealthy family's tomb in the second century AD and grew into a general burial ground. The main chamber is the strange part: Egyptian funerary scenes carved by craftsmen working in a Roman idiom, Anubis in a legionary's armour, bearded serpents with Greek shields. Nothing else shows the collision of the two cultures so plainly. It is damp, sometimes has water in the lowest level, and photography is generally not allowed.

Pompey's Pillar is up the hill above it, and has nothing to do with Pompey: a single 25 metre shaft of red Aswan granite raised for Diocletian in AD 297, standing on the ruins of the Serapeum, the temple complex that held a daughter library of the great one. Two granite sphinxes sit below it, and there are underground galleries where the sacred bulls and the book niches were.

In the afternoon, out to the tip of the Anfushi promontory for the Citadel of Qaitbay, which the Mamluk sultan built in 1477 on the exact spot where the Pharos lighthouse had stood, out of the stone that was left when it fell. That is as close to the lighthouse as anybody gets: blocks of it are in the walls and more of it is on the seabed just offshore. The fort is worth walking around, and the walk out along the breakwater with the whole curve of the corniche behind you is the picture everybody takes of this city.

The Graeco-Roman Museum holds what came out of the ground when modern Alexandria was built over the ancient one, mostly Ptolemaic and Roman: Tanagra figurines, mummy portraits, a hall of emperors, the sacred bull of the Serapeum. It was closed for restoration for years and has since reopened, and it is the single best place to understand what the city was.

Afterwards, walk downtown properly. The Eliyahu Hanavi Synagogue on Nabi Daniel Street is the survivor of a Jewish community that numbered tens of thousands before the 1950s and is now a handful of people; the present neo-Gothic building dates from 1850 and stands on a site used since the fourteenth century. It has been restored and can be visited on some mornings, with your passport, through a police checkpoint at the door.

That leaves the last hours for coffee. The old cafes around Saad Zaghloul have been serving since Alexandria was a Greek city as much as an Egyptian one, and they are the right place to sit before the train.