Yes you can.
To acquire Mew at the earliest point possible in the game, the player must not have defeated the Swimmer in Cerulean Gym and the Jr. Trainer (the one who says "I saw your feat from the grass" when the battle with him begins) on Route 24 in the grass. The player must defeat the Trainers on Nugget Bridge as usual.
The player must have an Abra that can Teleport in their party, and therefore, if playing Pokémon Yellow, must have reached Route 5 to catch one or traded for one beforehand.
The player must first heal at the Cerulean City Pokémon Center, so they may Teleport there later. With an Abra in their party, the player then needs to stand north of the Jr. Trainer so that he is just below the screen. It is recommended that the player now saves the game. The player must move down until seen and immediately press the "START" button. The player then needs to Teleport with Abra. If done correctly, the Jr. Trainer should get the indicative exclamation mark above his head immediately before the player Teleports. At this point, the menu will not work until after the player enters a battle, though it is imperative to not battle.
The player then needs to head east from the Pokémon Center into the Cerulean Gym and battle the Swimmer. After defeating the Swimmer (the menu should be available upon victory), the player then must head to Route 24, and after several steps the menu will appear. Upon closing of the menu, a battle will begin with a wild Mew.