Souk Wakif. On the main walking route, go to the very end (next stop car park area) and you can play backgammon in the coffee shop on your right after the place where the bedouin carpets are sold.
On the souk map in Marhaba they call it Eshairiq Coffee. Go upstairs, order a shisha and ask for a backgammon set. Nice view of the Fanar tower from there. Arabic coffee good too.
Enjoy your games!