Instructions
Travel to Cerulean City and travel across Nugget Bridge, which is to the north. After you pass the bridge, turn left and then south until you discover Cerulean Cave. Enter the cave and explore until you discover Mewtwo perched between two rocks.
Select a Pokemon that is not weak against Psychic-type Pokemon, such as a Bug-, Ghost- or Dark-type Pokemon. It will also be to your advantage to choose a Pokemon which has a "Sleep" attack. Save your game and approach Mewtwo.
Attack Mewtwo, avoiding powerful attacks that will knock it out. Choose attacks which do slight amounts of damage and use potions or cure abilities on your Pokemon after every few rounds. Use a sleep attack on Mewtwo whenever possible in order to avoid damaging your Pokemon. If Mewtwo has used its "Safeguard" ability in the previous round or two, you may not be able to put it to sleep.
Throw an Ultra Ball at Mewtwo when its health grows low. Switch out your Pokemon for a weaker Pokemon now that Mewtwo is weak. A Fighting- or Psychic-type Pokemon can be used for weak attacks that will gradually weaken Mewtwo. After each attack, throw another Ultra Ball until you successfully capture Mewtwo. Save your game after it is defeated.