The main highway from Aleppo to Damascus is in relatively good condition - I went up and down it last year (in a taxi), and remember it being pretty smooth and pothole-free.
It is very, very busy though. You also need to watch out for cars coming from the opposite direction driving in your lane - Syrian drivers have a tendency to drive in the centre of the road. You also need to watch out for livestock - this is probably one reason why Syrians often drive in the centre of the road. You'll see chickens, goats, sheep etc happily grazing on the verges of the highway. Oh, and everyone in Syria drives at about 160kmh. Apart from that, it's quite an easy drive ;-)
I've driven in Greece, Bosnia, Jordan, Romania, etc... and Syria is on a whole new level compared to those countries.
You may also find that doing this trip on a Friday is a bit easier - I certainly a found it a lot easier navigating Amman on a Friday than a weekday.
Seriously though, you should be fine, as long as you're very careful and very patient. And as Amagribi said, please never, ever drive at night in Syria, or anywhere else in the Levant for that matter. Night driving there is best left to the experienced, desperate or suicidal...