Requirements:
a. Abra
b. Make sure you haven't beaten Misty's gym yet.
c. Pokeballs.
How to do this:
1. Catch an Abra in the grass patches south of Cerulean City.
2. Go to the PokeCenter of Cerulean City.
3. Save your game.
4. Cross nugget bridge to the north and make a left.
5. Right when you get in view range of the trainer on the left side, press pause and teleport back to the Poke-center using Abra. You'll know it worked when you teleport, and the exclamation point above the enemy trainer appears, right before you fly back to the Poke-center.
6. Go to the Cerulean Gym (You shouldn't have gone to this gym yet) and immediately battle the swimmer, win.
7. Leave, and head back up Nugget Bridge. Your pause menu will come up. Then you will then be brought into battle with mew.
It appears you actually can get mew in Pokemon yellow, working and everything! I'm very excited to have gotten my game boy out to find out that this works! Very exciting! Good luck! Worked on my Game Boy Color Cartridge, no Problems!