My friends and I stayed here for two nights and would have gone back on our second visit, but unfortunately this was just after the bombing in Mombasa, and Britons were advised not to go into the city centre as the British Embassy had been evacuated due to a bomb threat.
The Princess Hotel has clean, fairly quiet double rooms for KSh800 (£7.50/$11) per night. Our room had the best shower I used during the whole of my four-month stay in Kenya (but I stayed in really cheap hotels). I'd recommend this place if you're on a tight budget, as they don't mind you sleeping more than two in a double room, although they contain two single beds and it's not very comfy.
Tom Mboya Street is very busy, central, and within easy walking distance of most places you might want to go, including banks and exchange bureaux, museums, shops, restaurants, and a couple of cinemas.
There is a bar and restaurant which are popular with the locals, and it gets very full when Premiership Football or Rugby matches are being shown on the TVs. I found the people using the bar to be very friendly, wanting to make conversation and generously offering out their stashes of miraa. Miraa is the shoot of a plant that when chewed is supposed to have a mild stimulant effect. It's legal (I think!), and they import it from Somalia. Try it, but you'll probably wish you hadn't--it tastes pretty horrid.