The expression “to jump the gun” is of racing origin, especially foot racing.
A contestant so keyed up as to spring from the starting mark a moment before the starter fires his pistol is said “to jump the gun.”
The expression is also used in hunting game, as when a pheasant “jumps the gun” by being startled and taking flight before the sportsman’s gun is in readiness. Hence, the phrase has acquired a general slang meaning of beginning a thing before preparations for it are in readiness.
In this figurative sense the usage is recent, probably not longer than within the past twenty-five or thirty years.