The Ornina was a stop at midday for us to regain strength. It is very modern and has been renovated, but retains the essence of a typical damasciana house, its courtyard, fountain, stairs to the first floor where you can eat or just have a drink, or smmoke a shisha. The list is quite complete, and the staff all very kind. Definitely a place to rest in a city that makes you very comfortable. It is close to the mosque, and situated in the Christian Quarter.
The first dinner I had in Damascus was in this restaurant, the Al-Khawali, near the main street (you can see it from the corner). It's a good place you can enjoy a typical Syrian meal, good desserts, and good atmosphere for not much money. Sometimes there's a line, so it's good to book ahead or go late in the evening since Syrian eating hours are rather earlier. You also can't miss heading up to the roof, especially at night, where you can see the spectacle of the city lit up at night.