There are 2 parts to this battle.  The first part you dodge his
attacks(shadow ball, bite, bombing, charge) then when he gets energy
from his life force orb, grind up the pipe coming out of his mouth
then use the homing attack( pres A in the air) and hit the orb on his
back.  Then when he uses the bombing attack that surrounds him, 
use the homing attack until you get to the orb and then hit it. 
Then when he lifts the gravity,  dodge the bombs and hit the orb
and kill him.  When you become SuperSonic and UltimateShadow, 
press A and B to attack and hit the swelling spot with the charge
attack.  If you run out of rings, you lose.  To switch
between Sonic and Shadow, Go under the Finalhazard(the boss) and it
will switch between characters.  When you beat it, you warp ARK
back into orbit and Shadow gets caught in The explosion of the boss
and dies.