Old English scot, sceot "a shot, a shooting, an act of shooting; that which is discharged in shooting, what is shot forth; darting, rapid motion," from Proto-Germanic *skutan (source also of Old Norse skutr, Old Frisian skete, Middle Dutch scote, German Schuß "a shot"), related to sceotan "to shoot," from PIE root *skeud- "to shoot, chase, throw."
Meaning "discharge of a bow, missile," also is from related Old English gesceot. Extended to other projectiles in Middle English, and to sports (hockey, basketball, etc.) 1868. Another original meaning, "payment" (perhaps literally "money thrown down") is preserved in scot-free. "Throwing down" might also have led to the meaning "a drink," first attested 1670s, the more precise meaning "small drink of straight liquor" by 1928 (shot glass is by 1955). Camera view sense is from 1958.
Sense of "hypodermic injection" first attested 1904; figurative phrase shot in the arm "stimulant" is by 1922. Meaning "try, attempt" is from 1756; sense of "remark meant to wound" is recorded from 1841. Meaning "an expert in shooting" is from 1780. To call the shots "control events, make decisions" is American English, 1922, perhaps from sport shooting. Shot in the dark "uninformed guess" is from 1885. Big shot "important person" is from 1861.
By the rude bridge that arched the flood,
Their flag to April’s breeze unfurled,
Here once the embattled farmers stood
And fired the shot heard round the world.
[Emerson, from "Concord Hymn"]