You'll have a great time in Egypt, and Cairo holds more than enough to fill a week; museum, pyramids, coptic area, citadel, old islamic area, the nile etc etc.

As for where to stay that depends on your budget.

The only downside is the incessant hassle you get from the locals. It almost ruined our holiday it was so intense. You'll never so "NO" so much in your life! One trick is to make out that you don't speak English.