Use your ATM card to get SYP and travel with the local currency. You will get the bank exchange rate and save yourself the hassle of carrying around cash. I have lived in Damascus for a year now and use ATMs all the time. Bank Audi has the most reliable ATMs depending on which card you use. I don't think bringing small USD or Euro notes is necessary as you can just use SYP everywhere and save yourself the hassle of getting bad exchange rates. I do keep some USD (100 dollar notes) and Euros on hand for emergencies though, as well as for smaller towns / day trips depending on how well you budget in advance when you withdraw SYP to carry with you. I personally use SYP for everything when I travel here.
I think cortex is right in saying that hotels vary in what they accept. Hotels can exchange money for you if you run out of SYP and cannot get to a bank to exchange money. Don't expect to use Visa much, depending on the places you stay and the type of travel you are interested in.
However, for last minute shopping, the Anat shop on Straight Street accepts Visa. Shops on the street give good prices in SYP and it is very easy to haggle here as there are almost no touts and very friendly shop keepers who are almost never pushy.