[OP09-013] - Yasopp (Emperors in the New World)

During my opponent's turn, I played "OP09-013 Yasopp" using this [On K.O.] effect, and activated its "[On Play] Up to 1 of your Leader gains +1000 power until the end of your opponent's next turn." effect. In this case, will my Leader gain +1000 power until the end of the current turn?

Yes, it will gain +1000 power until the end of the current turn. In this case, "until the end of your opponent's next turn" refers to the end of your opponent's current turn.