I have been living in Damascus for about 8 months now (I was living in NYC for the previous 4 years) and wanted to leave a post before leaving about some of the delicious food we've eaten here.
I think the best Arabic restaurant in Damascus has got to be the newcomer Narange. Located in the heart of the old city in a beautiful building with a terrace upstairs and a lovely dining room and open-ish kitchen. You can get all of the old standbys found in every decent restaurant in the Levant, as well as some more creative and seasonal dishes dishes. This is one of two restaurants that lives up to Western standards of service and cleanliness expected at a top-notch place. They serve a huge mezzeh which I highly recommend if you're going with a group. Alcohol is available.
My second favorite place is an Italian restaurant run by a totally adorable Syrian-Italian couple called Arabesque. I know it's not Arabic food, but when the idea of eating another stuffed grape leave and bowl of mettabbel makes me feel a little... unexcited, this is where we go. It's also in the heart of the old city in a delightful old house. They serve a small menu of delightful Italian dishes with ingrediants that you just can't find in Syria - gorgonzola cheese and great fresh fish, for example.
As for a good, classic Arabic meal in an old Arabic house, there is Bait Jebri, Al-Khawali and many many others in the old city. A place that serves amazing amazing food in a sort of sterile setting is called Mixed Grille, on the Tijara Corniche in the new city of Damascus. You know it's good b/c there isn't a single Westerner there and the men role in with a bottle of whiskey after 10 for dinner. A fun evening can be had by going there for dinner and then across the street to my favorite bar in Damascus (small, dim lighting, 99% Syrian crowd) called Kasebji. It has a red awning and is in a small house.
A great place to spend a morning is in a french-style cafe called Salle Sucre in Abu Romani. Great coffee, a shaded terrace, friendly staff, tasty pastries, sandwiches and salads. I hate to admit it but I also like going to Costa Coffee in the Four Seasons for an afternoon cup of the good stuff. If you go to Sahat Kaiffer Sousseh after 2pm in the afternoon and look for a donut shop/bakery on the circle, you will find some of the best donuts I've ever had. To the left of the donut shop is a guy who sells a GREAT shawarma. Don't be afraid to venture out of the old city.
If you're like me and you love wandering through food markets, the two best in the city are found outside of the Old City and are fabulous for wandering and tasting and learning: one is called Souq Serijeh just outside of the old city and the other is in an area up the mountain a little called Afief. Ask any taxi driver and they'll know both.
For those who aren't familiar with the classic Arabic food you must try while visiting: here is an attempt at listing out the basics (incomplete I'm sure):
Salads: Fatoush, Tabbouleh, Salad Sharqieh
Dips: Hummus, Hummus Beiruti (hummus with green onions), Hummus bi Lehma (hummus with meat), mettabel (smoked eggplant w/ tahini), baba ganoush (eggplant, tomatoes, etc), muhammara (my husband's fave - a sometimes spicy sometimes smoky walnut/red pepper paste)
Meat: Shish Taouk (chicken on a skewer), Kebab, Aleppo Kebab (made with a tasty tomato sauce), kibbeh (kibbeh is the quintessential Arab food - it's made of lamb pounded with bulgur and spices and served 100 different ways) - kibbeh meshwiyeh (fried), kibbeh naiyeh (raw), kibbeh maqlieh (grilled).