Home Run Contest is more or less the same as it always has been. Beat up Mr. Sandbag and then finish him off with a smash attack from the nearby Home Run Bat. You need to start the smash attack before the timer runs out, but you can finish it after time is up. If you're out of time and the sandbag is up in the air, try to aim for where you expect it to land. The sandbag will travel farther with more damage. Uncharged smash attacks do more damage per second than charged ones. Side and down smash attacks have a chance of sending the bag flying before you want it to, but up smash attacks are generally good for dealing damage without messing it up that way. This also allows you to grab the bat before you start attacking, which makes things simpler. The best way to deal a lot of damage in a short time varies from character to character, though.
If you're lucky, the sandbag will hit a trophy or custom part as it slides along the ground, but there's no skill involved with making this happen and it's just a happy bonus.
Distance is your only concern in this mode. It's not actually necessary to use the bat every time if you're playing as a bruiser. For example, one strategy for getting a high score that has been floating around is to use Ganondorf, start with three uncharged forward smash attacks (without moving from your starting position), which will knock the bag behind you. Then, still without moving, hit it with side attacks (not smashes) repeatedly until time is almost up. At the last moment, make sure you're on the left side of the bag, facing left. Do a Wizard Punch away from the bag, but turn and face the bag in the middle of the animation. (This makes it stronger than a standard Wizard Punch.) This should send the bag far enough to unlock the high score-related challenges for this mode, aside from the character-specific ones.
NEXT: Trophy Rush