"low in price, that may be bought at small cost," c. 1500, ultimately from Old English noun ceap "traffic, a purchase," from ceapian (v.) "to trade, buy and sell," probably from early Germanic borrowings of Latin caupo "petty tradesman, huckster, peddler," cauponari "to haggle" (see chapman). Compare, from the same borrowing, German kaufen "to buy," Old Norse kaupa "to bargain, barter," Gothic kaupon "to traffic, trade."
The sense evolution is from the noun meaning "a barter, a purchase" to "a purchase as rated by the buyer," hence the adjectival meaning "inexpensive," the main modern sense, via Middle English phrases such as god chep "favorable bargain" (12c., a translation of French a bon marché).
Sense of "lightly esteemed, common" is from 1590s (compare similar evolution of Latin vilis). The meaning "low in price" was represented in Old English by undeor, literally "un-dear" (but deop ceap, literally "deep cheap," meant "high price").
The word also was used in Old English for "market" (as in ceapdæg "market day"), a sense surviving in place names Cheapside, East Cheap, etc. To do something on the cheap "with very little expense" is from 1859. Cheap shot originally was U.S. football jargon for a head-on tackle; extended sense "unfair hit" in politics, etc. is by 1970.
German billig "cheap" is from Middle Low German billik, originally "fair, just," with a sense evolution via billiger preis "fair price," etc.
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"]