This can be earned at any point in the game, even after defeating the final boss. The Arrrrrcade cabinet is located inside of the hotel (the third building you can enter) where you purchase seeds for your garden.
There are five types of boards to choose from but I found Classic to be best, as you'll have more room to work with and no additional obstacles to overcome. Tiles are worth 50 points each when cleared in blocks of 4 or more of the same color. Sadly, there doesn't seem to be any sort of bonus points for clearing larger blocks or multiple blocks at once or in a cascade, so plan on getting rid of 500+ tiles. Furthermore, there's no incentive for stacking your tiles high in hopes on a massive cascade for extra points. With that said, make your matches as soon as possible and try to never have more than 1 or 2 rows of tiles at a time. The emptier the play space the better, particularly from about level 10 onward.
To achieve 25,000 points you'll need to reach level 16. At this stage the pieces fall almost instantly, but you can pause the game to give yourself time to determine the best placement (and breathe!) as well as plan for the next piece in line. I personally struggled through dozens of attempts where I consistently failed between 18-20k points, but the first try utilizing the pause button I just barely succeeded before panic set in. Those with better composure than I shouldn't have much trouble unlocking this achievement in far fewer attempts. Just abuse that pause button as soon as you feel your limits are being challenged.