I think the game you're referring to is the first Super Mario Bros on the NES (Nintendo Entertainment System). You know, the title that single-handedly revolutionised gaming
Yeah, Sunshine is the only "true" Mario game on the GameCube - the rest are just spinoffs that use Mario's name, not platformers. I would highly recommend the game to you, but you must know that Mario has evolved in his jump from 2D to 3D - it's a whole different experience. If you can, try out Mario 64 before Sunshine to get used to the new environment.
Also, Nintendo's next-generation system - the Revolution - will be able to download old games for you to play. So if you buy it you'll be able to play that old Mario in all its glory if you don't have the game for the NES right now..