1896, from Japanese, popular theater (as opposed to shadow puppet-plays or lyrical Noh dramas).
Kabuki comes from the verb 'kabuku', meaning 'to deviate from the normal manners and customs, to do something absurd.' Today kabuki is performed only by men, but the first kabuki performance was given in about 1603 by a girl, a shrine maiden of Kyoto named O-kuni, who 'deviated from the normal customs' by dressing as a man and entertaining the public with satirical dances in the grounds of the Kitano shrine. [Toshie M. Evans, "A Dictionary of Japanese Loanwords," 1997]
Alternative etymology [Barnhart, OED] is that it means literally "art of song and dance," from ka "song" + bu "dance" + ki "art, skill."