Although the Boomerang may be used in order to cut down the ropes to Puppet Ganon's tail, the common speedrun strategy (especially since the Boomerang is usually skipped) is to snipe the blue ball on his tail. In order to perform these snipes successfully, you should time your arrows based on the tail's movement. Shooting arrows when his tail starts to slow down can also be a successful strategy.
As soon as the 2nd phase starts, you can get an immediate shot right as Puppet Ganon goes up the room if you aim fast enough.
If you stand in a precise spot as PG goes down, you can prevent PG from going back all the way at the top of the room. The position you need to stand at depends on PG's rotation. If you did it correctly, when PG goes up, he wil get stuck below some collision and will drop faster the next time.
There is another quick strat you can do, which allows you to hit the Orb twice within one "phase". As soon as he falls down, you want to stand on the left side of his Orb and shoot it. After that you hold down-left on your controller and aim for the Orb while it is spinning. It can be tricky at first but you will get it down pretty quick.
There are three possible strategies for this phase. The first is to attempt to snipe his tail, which can be achieved consistently with good timing and practice. The other strategy is to run into Puppet Ganon's head with a bomb, causing him to stop, making it easy to hit his tail with a Light Arrow. Since Puppet Ganon's movement is random, it is sometimes difficult to hit Puppet Ganon with a bomb if he does a bad movement pattern. However, the first Hit is really consistent, because the way he moves at the beginning is always 1 of 3 patterns. 2 of these patterns can do a quick snipe in the exact same way. After this first hit, you want to make him stop to perform a mid-air snipe which can be set-up by standing straight behind Puppet Ganon.
A commonly discussed trick in The Wind Waker is the Puppet Ganon skip, a very hard and luckbased trick where you zombie hover up until you are above the entering to the final fight and rely on a bomb drop to kill a keese and spawn a heart to survive. Sounds difficult? That is because this trick is difficult.
There are several ways to skip Puppet Ganon, click here for more info.