One American dollar is worth around 46 Syrian Lira at the moment. You can exchange dollars at the airport, banks and exchange booths which are quite numerous in big cities. Most businesses will not exchange dollars, but might accept them as payment, though I highly recommend paying in Syrian currency only so as to get a better deal.
As for your second question, it all depends on how long you plan to stay in Syria. If it’s a short stay then I’d recommend bringing cash, if you plan to stay for a while bring an Atm card (provided you check with your bank first if it’s ok to use in Syria) or a Visa/Mastercard. Alternatively you can always use a money transfer service like western union if you ever run out of funds.