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Unfortunately, you are required to play through at least some of the main game in order to unlock the arcade game. The arcade itself is located in Frantic Factory, the third world, which requires 15 Golden Bananas to access (very doable and does not take that long at all considering each world has 25 Golden Bananas each). Once you access the factory, visit Cranky as DK to learn a special ability required to activate the arcade machine. You can then play as much of the arcade game as you want. Alternatively, if you photograph 6 Banana Fairies, you will unlock the ability to play the arcade game from the main menu screen, but this takes slightly longer.
You unfortunately need to get to the third world to unlock the Arcade game. To get there you need to fight both Armydillo and Dogadon or photograph a truckload of Fairies to get it in the Mystery Menu.
In addition to the pie factory level being absent, the physics are slightly different in the NES version from what I remember. The DK64 arcade game is the closest version to the original that I know of.