If you go via Riyadh, you have to cross the Nafud desert. In that case you have to take many camels along with your car. And there are no roads crossing the Nafud desert and unless you have GPS you will be lost.
But if you go to Dammam and cross Saudi keeping to its border with Kuwait and Iraq then eventually you will reach Arar and Turayf in KSA. From there you can cross the borfer to Jordan and then to Syria(hilly rds) and then reach Dimashq. From Damascus it is easy to go to Bayrout.
If you are really interested you can PM me and I can give you the contact no of the Sales manager of TISCO Qatar. You can consult him on your trip.
First go to TISCO Qatar(Teyseer Industrial Suppliers & Services Co. WLL) Instrument div. and purchase a trimble or Garmin GPS for the car.And then buy the map package for Middle East. Once you got this gadgets you can fix your itinerary.

Everything you need to know about living in Doha, Qatar