The shogi building is near the monorail station in Okinawa. Pick challenge 1 as it's the easiest game to win without moving your king. The objective is to take the opponent's king without moving your own king. It plays very similar to chess although shogi has a few extra elements to it like reusing your opponents captured pieces and powering up your own. Simply put you can ignore all that and focus on moving your pieces to take the opponent's king. You need to trap the AI in a corner with two pieces so you get a checkmate (it won't work with a single piece as they can always escape).
Refer to the video below for the exact moves to do if you need help: All credit goes to the original uploader of the video.