Hi Mo-Tech,
I can find out for you, I didn't need one because I am here on a Uk govt job. Intially I was given a temp permit for one month.Then had to get it registerd, what a saga that was, the authourities don't like bikes in Damascus (used as assinnation veichles in the bad old days) I have heard as a tourist that you can pay $70 which includes compulsory insurance for a months visit. You could include Syria on a carnet if you are visiting other countries in the region.
Despite its poor image I can recomend Syria, as a visitor with abike you be very welcomed, no rip offs, very freindly and helpful people.There are some premier sites in Syria and it is very under -touristed The roads outside the cities are good and have little traffic, petrol is about 60cents a litre and plenty of fuel stations. No real campsites but plenty of open desert (hotels are cheap) the general cost of living is very low. Security is not a problem and the crimerate is very low. Soon as I get the carnet info I will let you know, if you need more info drop me line.
Cheers
Duncan