Why do I get a sharp shooting pain radiating from my elbow up the back of my upper arm only when I lean on that elbow?

Ulnar nerve. You are leaning on and injuring your ulnar nerve or funny bone.
"funny bone" The ulnar nerve, which carries sensation down that side of your arm & hand, toward the 5th finger, passes across the elbow at the point we call the "funny bone." at that spot, it is very close to the surface, and when you lean on it at certain angles, it gets pinched. If changing the angle of your arm when you lean on it doesn't help or if it gets worse, see your doc to make sure that's all it is.