Microbuses across town cost 5-15 LS. A taxi across town should cost no more than about 50 LS.
If I had to guess, I'd say you could live quite comfortably in a shared two-bedroom apartment on an average budget of about 20,000 LS ($400 US) per month.
The tap water in Damascus is drinkable, so if you are living there you might as well drink it. Many houses have a separate drinking water tap in the kitchen which connects directly to the mains, while the ordinary cold water tap connects to a water tank and really really should not be drunk from (in a friend's house, we found the principal cockroach colony was centred around the house's cold water tank, which had an open top....)
Restaurants, as stated above, range from 100SP-200SP for the cheapest sit-down places, 200-400SP for the midrange places, and 500SP+ if you want something really fancy. If you want a break from Arabic cuisine (Arabic food is great and all, but after a month or two, this is likely), the a meal at one of the Chinese or Indian restaurants will probably set you back 400SP-600SP. There is also a pseudo-Mexican restaurant in Mezzah Villat which is just a bit more expensive.
Also factor in internet access, which can add up if you use it a lot (average 50SP/hour). And mobile phone service, which is crucial if you intend on having any sort of social life in Syria whatsoever. Anonymous
Of course, staying in one of these houses in Bab Touma probably means that if you are at all socially inclined you will end up being dragged out to moderately expensive restaurants or cafes at least every other night by the other residents, and potentially thereby spending far more than you would if you were staying in a cheap hotel. But this is a controllable risk.