By far the easiest way to play Mario Kart Arcade GP is on a soft-modded Wii. The Triforce arcade board which Mario Kart Arcade GP ran on was based on Nintendo's GameCube hardware which was also the basis of the Wii's hardware so being pretty much its native platform the game runs perfectly.
Soft-modding your Wii is perfectly legal and relatively easy to do, a good resource for this can be found at https://sites.google.com/site/completesg/
The best software to use to play Mario Kart Arcade GP on your soft-modded Wii is Nintendont which also supports other Triforce Arcade games including F-Zero AX, Mario Kart Arcade GP 1 & 2 & Virtua Striker 4.
The latest version of Nintendont can be downloaded from https://github.com/FIX94/Nintendont
Enjoy!
GAME CONTROLS Gas: R trigger Brake: L trigger Throw: A Insert Coin: Z