No, Paper Mario is not a playable fighter in Super Smash Bros. Melee (GameCube). He does not exist in the game’s code, and there are no cheats, codes, or unlock methods to add him.
Starters (14): Mario, Donkey Kong, Link, Samus, Yoshi, Kirby, Fox, Pikachu, Luigi, Captain Falcon, Ness, Jigglypuff, Peach, Bowser.
Unlockables (11): Dr. Mario, Falco, Ganondorf, Young Link, Mewtwo, Pichu, Ice Climbers, Marth, Roy, Mr. Game & Watch, and Sheik (transforms from Zelda).
Paper Mario made his playable debut as a DLC character in Super Smash Bros. for Wii U / 3DS (Smash 4) and returns in Super Smash Bros. Ultimate.