Early in the 20th century it used to be common for people feeling a bit run-down to go to the doctor to injected with a stimulant. By 1916 this remedy had led to a saying according to which a positive stimulation of almost any kind could be called “a real shot in the arm.”
We still use this expression in a wide variety of ways. It can refer to an increase of business in a company, a stimulus administered to the economy, to the hopes of a sports franchise or a politician running for office.
A simpler way of expressing the idea is to refer to a stimulus as a “boost.” Examples: “the flowers on my birthday gave my spirits a real boost,” “the large donation by the pharmaceutical company gave his campaign a major boost,” “the President is looking for ways to boost the economy.”
It’s easy to understand how these two expressions came to be confused with each other in the popular form “a boost in the arm.” After all, we go to the doctor for a booster shot. But the boost in this expression is a shove from underneath to raise the whole body, not a needle in the biceps. It makes more sense to stick with the traditional expression “a shot in the arm” or to simply use “boost.”