These are two really great historic sites that are wonderful to visit. Because they are so close to each other, you really don't need to book onto a tour and can visit by yourself.
At Pompey's Piller make sure you walk to the far corner of the site, as otherwise you will miss Cleopatra's Library which is an underground shrine dug almost underneath the Pillar itself.
At the Catacombs it is tempting to just stick to the duckboards, but you are allowed to wander off these and explore on your own - do it! The main attraction here though is the brilliant stonework that mix ancient Egyptian iconography with Roman images.
The sites are in a poor neighbourhood. Everyone is very friendly, but if this is your first time exploring on your own, the poverty might be a shock.

The best way to get to Pompey's Pillar is to just get a taxi off the street. Not all taxi drivers will recognise "Pompey's Pillar" so you will also need to ask for the area it is in, Carmus (pronounced Karr-Mooze). You can walk to the Catacombs from Pompey's Pillar - just take the road on your left from the entrance and it is about a 10 minute walk.

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For non-beach orientated things to do, the size and decoration of the Kom el-Shuqafa catacombs will remind you of a Spielberg film.

As for food, the Kadora (pronounced A-Dora) and the fish market offer some of the best seafood in Egypt.

A pleasant way to end a day's exploring is to take a calèch ride from near the Cecil Hotel, along the Western harbour, and then retrace your route on foot for a bite to eat at the fish market.

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If you can afford it, I would suggest splashing out and getting a room at the Sheraton Montazah.

The fact that it's on the edge of town might make it inconvenient for
some, but I found it eliminated most of the noise from the downtown
area, while still providing a major thoroughfare for catching a cab.

Its private, wonderfully under-populated beach, and a room with a balcony overlooking both the Montazah Palace gardens and the
Mediterranean were also welcome perks.

www.starwoodhotels.com/sheraton/property/overview/index.html?propertyID=438

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