I am very familiar with what you are asking and can confidently tell you that it is absolutely impossible to take a vehicle across the border that you do not directly own. Every car company will tell you NO VEHICALES ACPRSS THE BORDER!
You cannot even take another presons car. IT must be your own vehocle. The only way you will be able to crive is to purchase a car, obtain the necessary transit visas from Saudi ehich a local agancy does here, which theoretically could be done in 7 - 10 days, But you cannot even purchase if you dont have a license which you also need a resident permit for. SO it is completely unrealistic for you to drive to Dubai if you are just visiting for a short while. Also border corssings customs can be a hassle and take time. There is alot of crap to go through if your doing it the first time.
You have to go through customs, check you papers, buy car insurance for both Saudi and the UAE at the insurance kiosks stationed just before each border exit, so you can see it is not a smiplre manner if just pakcing the car and say "Lets go to Dubai today!" Alot of planning and logistics go into the drive for foreigners.
Better and safer to fly anyway and cheap fares do exist
with Emirates and Air Arabia to Sharjah. GOOD LUCK !