Travelling on budget? Just go to a youthhostel. There are several. For example Faisal youthhostel. When you get out of the old city through Damascus gate and cross the road, the hostel is at your left hand on the corner. Prices like 10/15 dollars-20 dollars per night. Depends on the room you choose.
Unique Qualities: Near the old city. It's just ten steps away.