I did the same before coming to Doha 4 years ago. You have to go to Saudi traffic department and ask for "Export" permit which is granted in the spot. You will have to hand over the number plate and they will give you a temp one for 7 days, which gives u enough time to travel to Qatar. At the borders, there is agents will do the rest for u for around 150 riyals. you can drive with these plates in Qatar till u pay the customes and clear the car status (5% of car value, and it's negiotable)
I wish the car deserves the headache
Good luck