Does the water take too long to drain or is the cat drinking the water as it’s exiting?
2 pieces of PVC or ABS with an elbow joint would solve this if the cat just wants the water while it’s running. You could just make an L with the pipe and have it exit right at the drain. Less than $20 bucks.
If the water is standing in the sink, you need a better drain and that’s not your job as a renter.
Also, is your cat okay? I had a cat that went bonkers for running water, poor guy had cancer.