My attempt to catch Mew for the first time. How to Catch Mew the Real Way:
Make sure you have a Master Ball(the 4th field multiplier ball) or at least the Ultra Ball(gold) before beating the Mewtwo Stage the 2nd time. Then you have about a 1/16 chance of having Mew appear if all three lights are lit or you get the Catch-Em bonus slot. You can only catch Mew with a Master Ball, so make sure that Staryu is open to upgrade your Ultra Ball to a Master Ball. It should take you about 10 attempts at catching Mew to keep your Master Ball long enough to catch him.