"currying favor," by 1946 (as arse-kissing), apparently from or popularized by military slang in World War II. Ass-kisser is by 1943. Grose's 1788 "Dictionary of the Vulgar Tongue" has "A kiss mine arse fellow," defined as "a sycophant." As for the verbal phrase, the Towneley Plays, c. 1460, has Cain telling off Abel with "Com kis myne ars." A 1668 book of "Songs Alamode, Composed by the most Refined Wits of this Age" has a song with the line "And thou maist kiss mine Arse," and of course the Miller's Tale.
In early 20c. American-English ass-licker may have been more common than ass-kisser, at least in print sources, perhaps from German influence. "Leck mich im Arsch" is at least 18c., the title of a Mozart party song, the phrase having been branded into the minds of German readers by Goethe in "Götz von Berlichingen" (1773), where it is the hero's dramatic reply to a call to surrender. (E.g. the reply of Tjaden to the bullying NCO in "All Quiet on the Western Front," where Remarque euphemistically describes it as "dem bekanntesten Klassikerzitat.")