There are a few other options that are much easier and cheaper than installing drywall if you are just looking to insulate. There are several products that you would just roll onto the ceiling and staple in place. My Uncle did his garage, by himself , so he could lift in there and he said it dropped the temp 20 degrees easy. The material is reflective on one side and kind of like a black mat on the other.
Drywall itself isn't going to help you a ton, unless you put insulation between it and the ceiling.
I would try the roll stuff first.
Here is an example, although I am not sure if this is the best brand, but it will give you a place to go from.
https://www.ecofoil.com/products/do...yFPQPfKCB4RmT1Mv8368Irh7Sh4DQsPQaAn1vEALw_wcB
Also, as a landlord I would be pissed if a tenant put up drywall without asking me, and I would 100% take it out of the deposit, but if a good paying tenant asked permission, I would probably offer to help with the bill, especially if they were doing the labor.
Good luck