Enjoy a typical Syrian meal
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.