While the word "shot" has many meanings; a gun being fired, small lead weights, projectiles in a shotgun shell and a loud noise, to name just a few.
When we use A Shot In The Arm we generally mean a "stimulus" of some kind.
The origin of this expression derives from the invigorating effect of injecting drugs. A "shot" is of course normal slang for an injection, either of a narcotic or medicinal drug.
That term has been in use since around the beginning of the 20th century, for example, this piece from the San Francisco Chronicle Supplement, October 1904:
"I varied hardly a minute each day in the time of taking my injection. My first shot was when I awoke in the morning."
'A shot in the arm' came soon afterwards and the first mention of a figurative use of it in print that can be found is from the Maine newspaper The Lewiston Evening Journal, January 1916:
"The veterns can give politics a shot in the arm and the political leaders realize it."
