To unlock the following cars in Car Attack mode, win the corresponding Grand Prix to get that car as an opponent in Car Attack mode. After defeating that car in Car Attack mode, you can then drive it in Car Attack mode:
CAR - GRAND PRIX 13th Racing Kid - Grand Prix 4 Age Solo S - Grand Prix 5 Assoluto Infinito - Grand Prix 5 Atomic Purple - Grand Prix 6 Burning Nightmare - Grand Prix 7 Dig Racing Team - Grand Prix 3 DK Team Racing - Grand Prix 6 Galaga RT Blue - Grand Prix 3 Galaga RT Red - Grand Prix 2 Galaxian Paradise - Grand Prix 4 Micro Mouse Mappy - Grand Prix 4 Racing Team Luigi - Grand Prix 8 Racing Team Peach - Grand Prix 5 RT Blue Mappy - Grand Prix 1 RT Bosconian - Grand Prix 2 RT Nebulasray - Grand Prix 2 RT Pink Mappy - Grand Prix 1 RT Xevious Green - Grand Prix 3 RT Xevious Red - Grand Prix 1 Terrazi Terrific - Grand Prix 6
After winning the above listed cars by finishing in 1st place in Grand Prix 1 through Grand Prix 8, additional difficulty settings will become unlocked. Win all the races on the Z (Platinum Cup) difficulty setting to unlock the following cars: 00-Agent, Caddy Car, Caddy Car, Galaga '88, Mario Racing, Pooka, Red Shirt Rage, and Shy Guy.