It takes two pulls on the tail to beat phase 1 of Argorok, it is possible to regrab his tail quickly the second time without having to go all the way to the top and wait for him. To do this after the first hit hold Up-Right on the analog stick, this will make you face towards the right pillar, then clawshot onto it quickly (can't be very low) and then clawshot him again quickly before he gets away
For the first hit of three is it quicker to void out and then climb the pillars than it is just to climb the pillars because Argorok is already waiting for you.
After Clawshotting up to the peahats just wait till Argorok starts to spit fire and put on IB. When you are below the flame clawshot further 3 times. This saves around 7 seconds if done on both cycles for Any%. By moving down on the clawshot chain you can speed this up slightly, as seen in the second video.
For his 3rd cycle Argorok will change the direction he breathes fire after the 5th peahat which is slower than the other cycles which only go one way for 7 peahats. However it is possible to cancel this flame going back by making the peahat you are on sink with Iron Boots, if you go low enough the game assumes you fell and hence stops his flame.
To do this get up to the peahats as soon as Argorok screams, the clawshot further three more times. Now wait till Argoroks flame hits the peahat next to you and put on Iron Boots to sink below the flame. As soon as Argorok starts breathing fire again, unequip Iron Boots and clawshot onto his back when he stops. Now finish him off with Finishing Blow.
In 100% it is possible to skip the final cycle by using Fairy Tears or Rare Chu Jelly.
There are two methods for 2 Cycle Argorok, the original being harder as you have to clawshot to Argorok without aiming because you have very little time. However it has been improved so we can skip a peahat therefore making the timing easier so you can aim, however it does require some what precise timing on the drink animation but this can be buffered easier so this method is highly recommended.
Below is the improved method to skip a peahat and makes the strategy easier. It is important to note that the pillar you use seems to matter due to peahat positions, so you should try and copy the video exactly. The pillar stood on is on the left side at the back if you were looking at it from the vines where you enter the fight.
Below is the original method that requires the clawshot to Argorok without aimming (Harder)
A video tutorial about the boss fight, with commentary.