Well I think the answer is yes you can drive there. I am basing this on the fact that you can get a car and driver from Beirut to either Damscus or Amman. I'd suggest contacting the hotel directly.

A quick check on Google came up with a lot of people describing their journeys by Service Taxi (shared old mercedes), but I think you are after something more stylish than that.

As for heading onto Petra, that is perhaps a little too far to do in one day from Damascus. I'd suggest an overnight in Amman, then take a locally sourced excursion from there. Petra seems to be well worth 2 days anyway.