mid-15c., "now, present, of the moment, current," from Old French instant "near, imminent, immediate, at hand; urgent, assiduous" (14c.) and directly from Medieval Latin instantem (nominative instans), in classical Latin "present, pressing, urgent," literally "standing near," present participle of instare "to urge, to stand near, be present (to urge one's case)," from in- "in" (from PIE root *en "in") + stare "to stand," from PIE root *sta- "to stand, make or be firm."
Sense of "immediate, done or occurring at once" is from 1590s. Of processed foods, by 1912; instant coffee is from 1915. Televised sports instant replay attested by 1965. Instant messaging attested by 1994. The word was used 18c.-19c. in dating of correspondence, meaning "the current month," often abbreviated inst. Thus 16th inst. means "sixteenth of the current month" (see ultimo).