I'm male, but I think that for most places in Cairo you should be fine on your own at least during the day. I walked from the Nile Hilton to the Cairo Museum (amongst other places) on my own and it was just like walking in most any crowded 3rd world city. Lots of people all shapes and sizes, men and women, walking about.

The main thing to worry about in Cairo is being "taken for a ride" by extremely high-pressure salesmen disguised as taxi drivers. Make sure your hotel arranges any taxis for you and tell them you do not want to listen to any sales tactics. You have to be firm even if you feel it is being rude.

Gauge the scene on your own for walking around at night. I probably wouldn't recommend it for a woman, but it depends on what area you are in. If you are in a good hotel, talk to your concierge. Maybe you'll meet a nice couple and go out with them.