Hi,
I moved here at the end of January to study Arabic. Finding a place involved a lot of searching ie looking at places and took some time. I know some people who can help you find a place if you'd like their details.
I've rented an a niceish (it's good but not perfect) 1 bedroom apartment in a place called shaalan, which is about 30 minutes walk both to the university and to the old city (and north of each). this costs around 500USD per month.
Many students go for a room in a Damascene house, with shared facilities, in say Bab Touma or Souq Saroujah and these are between 200USD and 400USD (tops) per month, depending on the size of the room and how much stuff you share (some have their own bathroom)
You can of course also rent an apartment with other students and it works out cheaper than just for yourself.
Let me know if you want more info, or if you'd like this in Syrian pounds - I'm going from Syrian pounds to Aussie dollars to US dollars above!