In Dobuita, there is a games arcade.
Hang On is the motorcycle that when you ride you can choose to enter an amount of money. (+ to add credits, - to subtract credits, up to put to max)
Press X when you're ready to play and press Start to play a game of Hang On.
You should get the trophy on the menu screen.
But if you want to play, R2 to accelerate and left stick to control and lean.