The specialist with the most training to care for the surgery around the ulnar nerve is a hand surgeon.
Other possibilities are: general orthopedic surgeon, plastic surgeon, and a neurosurgeon.
If the ulnar nerve is being compressed by an anatomical structure, there is not really anything that can be done at home. The compression needs to be relieved.
But, if the problem is more positional, then avoiding putting the nerve under compression will help. So, at night, avoid flexing the elbows up significantly. It is hard to prevent with when you are asleep, or you might try soft elbow braces. Most patients do not tolerate stiff or hard elbow braces when they are sleeping. So, soft neoprene sleeves or sports elbow pads will sometimes keep you from bending the elbows up too much. Another trick is to take several old tube socks, cut the toes out, and put them over the elbow. All you really need is something the keeps you from sharply flexing the elbow (up past 120 degrees).
The same is true with the wrist. Some wrist supports, just to keep you from flexing the wrist sharply.
Try not to rest your chin or head on your hands. This can put pressure on the ulnar nerve at both the cubital tunnel and Guyon's canal.
Some chiropractors say that you can stretch the wrist and elbow to prevent or delay surgery. You might look into that also.
If the compression is coming from an inflammatory condition, then treatment of the inflammation will help to reduce swelling, and thus compression on the nerve.