Private flying around Dammam

I don't think so. 

Pretty sure the closest microlight or ultralight you would find would be in the UAE.

Surely there are aviation websites that would detail such rental facilities.

Well, I can live without women and booze for a while but this is too much!

Between the security issues and the fact that civil aviation is not part of the ME culture (largely because of security issues, but also education and finances), you will have to get used to doing without this, as well...or head on back home.

hm, somehow neither of these arguments make sense, eve though the security argument was used as an excuse to snuff out private flying in Egypt after 9/11
Of course, nobody mentions that the 9/11 pilots learned to fly in the U.S.

Try Wallan aviation. They are a licensed Cessna training center. I don't know how desperate you are for flight time, but the cost per hour is ridiculously expensive here. Apparently they have to import the avgas, and the cost is something along the lines of 400 USD per hour...