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.
In this Phase you just hit his Orb intentionally. However, there is a 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.
There are several ways to skip Puppet Ganon. Click here to see them all.