It's not very hard to find rides with truckers from Istanbul to Gaziantep, you just have to find the right spot.
guaka did it in 2008 but doesn't remember much of the details. Maybe he even took a bus to Kilis. It looks like you have to go towards the airport. There's not much traffic south from Kilis (on the Gaziantep Kilis Yolu), so it's probably better to first get into the center of Kilis and then go South from there if you're heading to Syria.