There is no password for the Omega Ruins, it is found manually on the Spira Search option from the Airship Main screen. Depending on the version, Omega is either a pushover, (INT) or...
even more of a pushover (NA).

He's insanely easy, as nearly none of his attacks do damage against all characters, and when he DOES use it, it is once out of ever many turns.

In the American version, you can beat him right after beating the first encounters of Sin on the Airship. He's easier than his cronies, and Ultima Weapon had a better drop. (Bribe 1.4 million for 99 Pendulums)