Don't open the power supply, even unplugged there are caps in there that can be dangerous.
I'd trash the power supply, since a short can damage other more expensive components. I'd clean the case and case fans with warm water and lint-free paper towel then let them air dry thoroughly.
Furthermore, I'd wait for any circuit boards to dry completely from urine, try to blow off as much urine residue as possible with air pressure (e.g., Disposable Gas Duster), then try to dry-rub as much of the remaining with high quality shop towel and Q-Tips. You can use a tiny amount of distilled water on the towel to increase transfer, but less is more (talking droplets, or the stuff from your breath, not saturated).
If you run it while wet, it may fry things, if you run it after it dries it is more likely to just smell really terrible for a while.