In the Wild West booth, you will come to the Creaky Canyon Alley Ball Challenge mini game, something related to the arcade version of Skee-ball.
To get access to the mineshaft you will have to shoot the ball to the far left, do this five times. . Easy way to get this is to line up your ball in the middle and slightly adjust your angle to the left. Use the RIGHT STICK to adjust your aim & the RIGHT TRIGGER to shoot the ball.