Hilton Alexandria Green Plaza

The 5-star hotel inside the...

The 5-star hotel inside the extensive Montazah Palace grounds is said to be good. There is a Sheraton a long way out, also near Montazah. The Cecil Hotel in central Alex is historic etc (Churchill stayed there, as did the other Allied top brass in WW2), but seemed to me as a casual visitor (I had a coffee there) a bit dead, although that was off-season, in January-- and during Ramadan as well.
The tiny hotels listed in Lonely Planet and other guides, quite a few of which occupy floors or part-floors of old commercial buildings in the next street westward after the Cecil, as you go along the Corniche, are reasonably (some, very) clean and OK and charge as little as Egyptian Pounds 6 per person per night (less than US$2). They are mostly several floors above ground level and the entrances are usually open, dilapidated atria or hallways. They are better than the first impression of the public areas would have you think!
I would stay at the San Stefano on the Corniche in the Chatby area if I were in Alex again and had (educated guess) about $40 to spend: romantic little peninsula location, just about walking distance to central Alex along the Corniche and near the Rushdie area, which is good for consulates and interest walks.If you insist on renting an apartment, one in a good central-ish area might cost as little as $150 a month, off-season (much more in the summer). I paid the quivalent of $150 for a month in the supposedly swish gated beachfront suburb of Mamoura Beach. In summer, this is Egypt's equivalent of Venice Beach, California, but in Alexandria's sometimes gloomy, sometimes wet winter, at least, you would do better to stay in central Alex, as the depressing picture below of my road in Mamoura (January) shows!It was not all like that: Mamoura Beach main road, in winter sunshine
The huge mosque near my apartment block, well within earshot of the hideously amplified hazan (call to prayer), starting at an unbelievably early hour!