A mix of wild camping and hotels is the norm for most cyclists in Syria. Prices certainly permit a fair bit of credit card touring but hotels aren’t always placed at close enough distances to allow you to leave the tent behind entirely unless you want to ask local families for a place when no hotel materialises. 
A typical budget hotel will have double rooms starting at about 500 SP per night, usually with shared facilities. You can expect a range of accomodation in the bigger towns from budget to boutique hotels and an increasing number of international five-star brands like the Sheraton and Four Seasons in Aleppo and Damascus. Prices can rise dramatically in the summer, when citizens from the Gulf states come to enjoy the relatively cooler temperatures, and during high tourist seasons such as around Easter and key festivals like Eid.
There are very few official campsites in Syria. Two can be found about halfway between the border post of Bab-al-Hawa and Aleppo, one on the outskirts of Hama and another about 5km from the centre of Damascus. In general, wild camping will be the way to go and normally there is no problem finding a hidden spot to pitch your tent. You can also approach families for permission to camp on their land, which almost always leads to an invitation to spend the night in their home. Asking at military or police posts for permission to put a tent nearby is another option with the added benefit of extra security and water on tap.
If you plan to wild camp, it is a good idea to carry a letter explaining a little about yourself and your trip, which you can get translated into Arabic by anyone you meet who speaks English (there are many who do). This will help explain what you need when you approach people asking to camp or when, as inevitably happens, a shepherd approaches your tent to see who the visitor is.