Can red and blue version catch mew?

Answer 1

You can't get Mew legitimately in Blueor Red unless Nintendo have given it to you. There is a glitch, which does work and I can vouch for it, but I haven't tested whether Mew learns the moves properly. If it's just to complete your Pokédex then I suggest you could give it a try. You can't use it until you have the Fly HM.

You mustn't battle the trainer near Nugget Bridge (in Cerulean) who has the level 17 Slowpoke, or the Gambler on Route 8 (Left of Lavender Town) with two Poliwags and a Poliwhirl. Also, I'd suggest you have a couple of low-level Pokémon on your team and some decent balls.

North of the trainer on Route 8, there is a building. Walk to the door and do not go inside. Take one step down and immediately press start and fly to Cerulean. When you fly away, the gambler will "see" you, but you won't enter a battle.

Go to the trainer near Nugget Bridge and battle him (make sure your team can beat his Slowpoke). When you battle him, you must make sure he walks to you, otherwise the game will freeze. When you've beaten him, fly back to Lavender Town and got west to Route 8. When you enter the route, your start menu will come up. Press B to exit it, and you will enter a battle with a wild Mew at level 7. It can be difficult to capture. As long as you didn't save at any time when you're doing this, you can always turn off your game and try again if you failed to do it right.