Not sure about the drive from Damascus, but assume it can be done.

I was in Amman a year ago today, spent two weeks working in the embassy there.

Petra is a must see, plan on at least two nights there. This is a brief description of my trip, taken with colleagues so we had a car;

Drive down the main desert highway, got to Wadi Mousa the first day. That's a town a short walk from Petra. Stayed at the Movenpick hotel, quite nice and the closest to the entrance to Petra. Went into Petra first thing in the a.m. spent the whole day there. Won't describe Petra, others have done it better. Check Wikipedia.

On leaving Petra we went to the Dead Sea and drove up the coase, visiting Mt. Nebo on the way.

Don't have much time at the moment, but no-one should miss a chance to see Petra.