Take the train. Trains run at least once per hour (this is probably the busiest section in Egypt), the trip is a bit over two hours if I recall correctly, and cost is minimal (less than $10) even in the reasonably comfortable air-conditioned first class. The train station in Alex is within easy walking distance of the central seafront, although the town itself is widely spread out over the coast. You can trundle along the coast with (ridiculously cheap) tram if so inclined.

However, be warned that modern-day Alex is not an "appealing" town in, well, any sense of the word. For most part it's a huge overgrown slum that has long since lost its famed international character thanks to Nasser's nationalization campaign, which drove out all non-Egyptians. If you know your history well and have read Durrell's Alexandria Quartet you can still find a couple of spots with character though, notably Cafe Trianon and the Elite Restaurant. The other things worth looking at are probably the fortress of Qait Bey and of course the recently recreated and finally opened Great Library of Alexandria.