It is recommended that you save-and-reload when attempting this trophy. Depending on your reflexes, hitting a home run can be tricky. At least for the easiest difficulty, the method I tried was aiming the bat so that the top half of the battle would overlap the bottom half of the ball. After that, it comes down to just learning the timing.
While the reward for spending time at the batting cages is minimal, I do not advise reloading a save after doing this once to regain your day activity, as you the game to know you have done the batting cages once in order to purchase the book Batting Science in Shinjuku.
All credit to xNinjaOfChaosx