Start by having Zelda walk a short path so she won't follow Link around. For a while, Byrne will leap around on the pillars, shooting energy at you. Avoid these energy beams.
Eventually, Byrne will target you with his mechanical arm. You'll see a purple target on Link. If the arm hits you, it will hurt you. If it hits Zelda, she will be stuck and you must hit the arm to knock it away.
Instead of letting this happen, get away from the Phantom and allow the arm to hit the arena floor instead of you. It will get stuck on the floor, and Zelda will say that she can grab onto it. Use the stylus to guide Zelda over to the hand. When she grabs it, she will pull on it to bring Byrne down to the arena floor. At this point, Link can slash at him with the sword.
If Byrne charges up an attack while up on the poles, have Zelda move around. If it hits Zelda, it will stun her. Hit Zelda with a weapon to get her to stand back up.
If Byrne jumps down to the floor, hit him and keep him away from Zelda. Have Zelda walk to the middle of the arena. When Byrne charges up some purple energy, run around behind Zelda so that she is between you and Byrne. Then Byrne will jump down and Zelda will grab onto him. While Byrne is being held, have Link go hit him with the sword.
Repeat this process until Byrne is defeated.
After the cutscene, go up through the north exit and then go up the spiral staircase.
Watch what happens. Anjean will tell you to go to the Sand Temple to get the Bow of Light. She will add tracks to the desert so you can reach it.