Remember that Golf Story only shows the point of first impact for the ball, and not even adjusted for wind and your QTE skills. Once the ball comes down, the slope of the ground also plays an important role in how the ball will continue.
If you need the ball to stop as quickly as possible, after pressing Y and adjusting your reach with up/down (and ZL, ZR or both if it's a short approach shot or long putt), press X, move the dot to the bottom of the ball, then also press R so that the shadow is on the bottom of the ball while confirming with A. This is the maximum possible amount of backspin you can add to the ball. Be aware that the ball will now travel very high into the air, so wind will influence it quite a bit more.
Anyway, with that, the ball will come to a stop as soon as it can, possibly (depending on your stats) even rolling backwards.
The material the ball will hit drastically affects how much it will bounce. Pavement or green will let it hop along without worry, while deep rough or bushes will immediately slow it down, so aim your target marker for the first impact to land in those if at all possible. Also consider aiming for a treetop - the prediction line will accurately show how your ball will react.
Most of all, don't be too greedy. Space your shots so that you can use a wedge
well below 150 yards on the last shot to the green. If you have trouble with overshooting, try to go for short distances/use the approach shot function with the Z triggers to impact as little forward momentum to the ball in the free range mode to get a feel for it. Finally: Invest into some other stats than power.
The in-game explanations for the actual controls and golf mechanics are plain horrible.