The "funny bone" is not really a bone. It is simply the ulnar nerve as it passes across the elbow. It's also not nerve "endings" but a major bundle of nerves, mostly sensory nerve fibers coming back from the hand and headed toward the armpit and then the neck and then the brain. Do you notice that the pain you feel seems to be shooting down your forearm to the the side of the hand opposite the thumb? That's because the nerve fibers in the ulnar nerve are sending their signals up the nerve. When the whole nerve is stimulated the brain feels the pain in the forearm and hand.
In most locations where nerves are close to the surface they run in grooves in bone which serve to minimize the chances of a blow injuring the nerve. If you feel on your face, along the undersurface of your eyebrow you can feel one of these grooves, and if you press on it, you will find that it is quite sensitive. That is the supraorbital nerve coming out to the forehead through the supraorbital groove.
There are other areas on the body where nerve bundles are close to the surface or can be felt. There is a sensory nerve bundle running just in front of the anterior-superior iliac spine that can get pinched and causes a painful conditon called meralgia paresthetica.
If you place your fingers in the front part of the armpit you may be able to feel an arterial pulse there. It is where I learned in Boy Scouts to apply pressure to control serious bleeding (although the right way to control bleeding is almost always direct pressure on the wound.) There are nerves running down to the arm right next to that artery. That is not a typical location for nerve injury.
The saphenous nerve runs down the inside of the leg and if you poke around just behind the knee toward the inside you can probably find some tender points that are the nerve's location. This picture may help.
Doctors need to know where the nerves are close to the surface so they can evaluate lacerations or administer anesthetic nerve blocks such as in these pictures of nerves in the wrist.
I found some anatomy lesson videos I thought were pretty neat, but be advised, these are videos of real dissections on preserved human bodies. Not for the faint of heart but who knows, you could be a future orthopedist. To see the videos visit the Dartmouth Medical School Human Anatomy Learning website and click around looking for "Overview Video".
There are 3 types of nerve injuries. Neurapraxia is the mildest form and occurs just like you describe: the typical "funny bone" sensation. Axonotmesis is the second most severe and causes a much more lasting effect, although it usually goes away in weeks to months. Neurotmesisis the most severe and often permanent. It results in damages to the whole nerve, including the central part called the body. Any nerve, if hit directly, can exhibit any of these signs.
Why does the funny bone (ulnar nerve) problem happen so often? Simple. It's very large and easy to hit on things. When you hit it, on something hard, it gets squished between the surface you hit and the bone directly underneath. There isn't anything to help cushion it.