Qasr Tuba can be reached as a one day trip from Amman (its takes about 1.5 hours one way) and you don’t need a four-wheel drive to get there.
One possible route to reach the castle is: head south from Amman on the Desert Highway, and just before Swaqa prison there is a road to the east sign-posted as Qasr Tuba and Hazardous Waste Disposal. Follow the asphalt road until the fenced Hazardous Waste compound. The asphalt finishes abruptly and you should take the first visible track on your left.
Alternatively, you can take the Azraq-Jafr road south of Azraq, and turn right just after the Tuba police station (the road is not asphalted).
Another possible route is to take a 4×4 track from Qasr Kharaneh to the south – aim for the Tuba GPS point in the file below.
The kmz file with points is here.