One of the best budget places to stay in Damascus, the hotel inside the modern St. Elias Monastery, part of the Greek Orthodox Patriarchate of Antioch and All the East, comes recommended by many travellers for its welcoming atmosphere, 
friendly staff, cleanliness, reasonable prices, hot water (possibly limited to certain hours) - and especially the effective heating in winter. As it's a monastery, the gates close at midnight when you need to be inside. Find it 1km or 15 minutes walk south of the old town's eastern Bab Sharqi city gate. Single rooms $20, doubles $25.