Cairo is fine, even for a single woman.
I never once felt unsafe there, even walking around late into the night, by myself.
I stayed at the Sara Inn Hostel and highly recommend it. It was near the museum.

I have said before that hostels are not for everyone but I love them. You get to meet some great people there.
I could have had a hotel with an airline discount in Cairo for about $100 but I had my hostel for $8.
For me, a bed is a bed. I am there to see the city anyway and not sit in the room.
I have stayed in a hostel about 10 times and never did get bed bugs, nor did I experience any other issues.