Mew glitch. Defeat nugget bridge, DO NOT ENCOUNTER THE TRAINER LOOKING NORTH ON THE GRASS WHERE ABRA SPAWNS, catch an Abra, be sure to visit the pokemon center in cerulean city before this next part. Line up directly north of the trainer by the Abra spawn, just one block out of his line of sight. Press down and start at the same time. If you did it right, the start menu opens and the trainer is visible in the bottom of the screen. Use teleport on Abra, returning to cerulean city. Go into the gym, defeat the first swimmer, then return to nugget bridge. Upon entering the route, a battle should start and that'll be your mew. Catch it.

It's real, but it's yet another gen1 memory corruption glitch that reliably overwrites the Pokémon encounter table if you do it exactly right. If you search for "Mew glitch" you'll find a bunch of videos on it. It involves some very precise steps and specific battles on that one bridge where you get a golden nugget, IIRC.

I don't recall it exactly but its something along the lines of lose a battle to a particular slowpoke, travel to that patch of grass next to nugget bridge and use the escape rope or teleport trick when starting the battle with the trainer there, head to lavander town and head west. Once you leave the city you run into a wild Mew.

You could look it up just as easily as I could. I don't have it memorized