No you can only do it if you have a cheating device like an Action Replay for the Game Boy Advance for example since distributions in which you would've been able to get Mew are no longer available for the Game Boy Pokemon games.
No you can't you can trade one from pokemon emarald, where you can catch one on the faraway island.
source: http://serebii.net/pokedex-rs/151.shtml
You can't get mew in firered or leafgreen.
Its not possible to get mew in firered only emerald can via Nintendo event.
You can't catch it on Firered you can only catch it in Emerald
Cerulean cave
Pokemon firered emerald
Nope, you have to trade him.
The Cerulean Cave
sadly,no
can't but you can get one from emerald if you have the old sea chart and catch mew at faraway island.
No, you to migrate it from Pokemon firered/leaf green.
Unfortunately, Mew is not a catchable Pokemon in FireRed. It is impossible to catch it without a Cheating Device. Sorry, The interwebs fooled me, too.
Put it to sleep then use an ultra ball.