No idea where else you have been in the GCC, but KSA is it's own animal. You are going to be living South of Riyadh by over an hour on the highway (excluding driving through the city when you arrive). From what I understand, Al-Kharj has a lot of government developments going on in that area at the moment as there are a few historical areas in that city. You will have no choice but to live in Al-Kharj. I'm assuming you have done everything through your headhunter or liaison when it comes to your living situation, right? All details go through your headhunter/liaison; I can't answer those for you.

If your office is located in Al-Kharj, you need to try and live as close as possible. Traffic will still be insane in Al-Kharj, even though it's not Riyadh insanity... You don't need a car at all, unless you want to get one down the road. Just use any of the rideshare apps to get around. That said, if you are working far off somewhere outside of the city (i.e. building site, industrial area, etc.), then you might need to talk with your company on your options. Then again, I don't know the details around your contract and how long you plan on living/working in KSA.