When you first encounter the Omega, it will swipe Samus, forcing her back into a corner, extremely weakened and paralyzed. The Core-X will then appear, transform back into the SA-X, and begin attacking it. The SA-X is easily killed by the Omega's claws, allowing Samus to absorb the Core-X into her body, making her whole again.
The battle from then on out is quite easy. Angle your newly-acquired Ice Beam diagonally upward, and shoot into the Omega's weakpoint: its glowing pink chest. This will push the Omega backward. When it reaches the right side, the Omega will move forward and raise up its claws in anticipation to swipe at you. Use this moment to move towards the left side of the screen to easily avoid its swipes, and then run back to the right and keep shooting. Keeping with this back-and-forth movement allows the Omega to be painlessly defeated.
Do this all before the time runs out, and the game is won!