It takes two pulls on the tail to complete Phase 1, but it is possible to regrab the tail quickly the second time without having to go all the way to the top and wait for Argorok. After the first pull, turn Link to face the pillar, then quickly clawshot the lowest panel of the pillar (not precise, but avoid the very bottom of the clawshottable area). Once Link is on the pillar, clawshot the tail again to end Phase 1.
The direction Link is facing after the first pull's cutscene depends on the camera angle during the pull. If you use the c-stick during the pull to align the camera the same way every time, you'll be able to hold the same control stick direction to turn Link around every time.
For the first cycle, it is quicker to void out and then climb the pillars, skipping Argorok's flight around the arena, than it is to climb the pillars without voiding out.
When clawshotting back onto a pillar after falling with Argorok at the end of a cycle, it is possible to ascend too high too soon and cause Argorok to roar before Link has reached the top of a pillar. To avoid this, wait to clawshot onto a pillar from the ground until the wind has visibly begun to blow quickly.
During the 3rd cycle, Argorok breathes fire in one direction for 5 peahats, then turns around and breathes fire in the other direction. This takes longer than the other cycles, in which the flame goes one direction for 7 peahats. It is possible to cancel the second half of cycle 3's flame by sinking so low with the Iron Boots that Argorok acts as though Link has fallen to the ground
The optimizations shown and described below can save up to 11s over the standard fight described above, but early roars (cycles 2 & 3) are difficult and somewhat risky. Just using the second peahat flame cancel strategy in cycles 1 & 2 saves 4s and is low-risk.
(This paragraph describes the ideal beginning of the cycle, but ultimately the goal is just to claw up the pillars twice asap.) Roll 3x towards pillar and claw up high enough that the next claw will put Link in range of the peahats. Then, ideally, put on boots and immediately claw to about the level of the highest thick horizontal line on the other pillar, which is about as low as Link can target the peahats from; this will make Argorok roar before Link gets off the pillar even if you claw to the first peahat immediately, which is ideal. If you claw higher than the line or take a moment before clawing, Argorok has more time to descend and turn towards Link's position on the first pillar, so it then takes longer to ascend and turn to his new position on the second pillar before roaring.
Shoot the claw towards the first peahat before Argorok roars but close enough to Argorok reaching Link's position/angle that it still roars before Link leaves the pillar. Clawshot to the second peahat right away. Descend for a flame cancel. Because Argorok doesn't turn between roaring and shooting flame, it will not have time to turn and face Link, so from here, you'll only need to clawshot to two more peahats before clawshotting Argorok instead of three peahats as is needed if the flame cancel is done from the first peahat, saving about 2s.
This can be done exactly the same as cycle 1 to save another 2s.
It is faster by about 3s to go for an early roar. To do this, dismount Argorok and roll 3x towards a pillar behind Argorok's back. Immediately aim and claw up; the wind gust will be opposite the direction you're clawing and will start shortly before the claw reaches the pillar. This gets Link above Argorok as it rises to the top, which makes it stop and turn to face him and roar.
Put on boots and claw up to the second pillar ASAP; it doesn't have to be instant, but pretty close to it. Tap L twice to target the peahat that isn't right above the flame, then claw up to it immediately. Descend to flame cancel (look for flame appearing on screen), then clawshot across the peahats to get to Argorok's back. Sometimes this takes 2 peahats and sometimes 3 for unknown reasons.
If you fail to claw to the second pillar quickly enough, Argorok will start the flame very low, and when it rises too much while spewing flame, it will stop spewing and reset the cycle without being stunned. The best backup in this scenario is to clawshot back down to the bottom of a pillar or drop to the ground, then clawshot back up.
It is even faster to do as described above but go to a pillar in front of Argorok after dismounting instead of behind it, since then it won't need to turn before roaring, but this requires such fast aiming that it may not even be possible on Gamecube; it is possible on Wii, but is very difficult.
An early roar also speeds this cycle up. It is exactly the same as described above for cycle 2, except you don't need to put the boots on and also don't need to double target the first peahat, as either peahat direction will work. After getting to the fourth peahat, put on boots and descend to avoid the flame. After the camera shifts slightly indicating Link is low enough, take off the boots like in the normal 3-cycle strat and wait for Argorok to stop spewing flame, then clawshot on.
It is possible to skip the final cycle of Phase 2 by using Fairy Tears or Rare Chu Jelly.
There are two methods for 2-cycling Argorok. The original is harder, as it requires clawshotting onto Argorok without aiming for lack of time. The newer method gives the player time to aim by requiring clawshotting to fewer peahats at the cost of somewhat precise timing on sidehop and bottle inputs.
This setup will not work if begun on the wrong pillar in the arena, so use the same pillars shown in the video.
This video shows the entirety of phase 2 to make it clear which pillars to use for the trick.
Original instructional video:
A video tutorial about the boss fight with commentary:
In cycle 1 of Phase 1, clawshotting onto Argorok's tail on the first possible frame will softlock the game, and Argorok will fly Link around the arena indefinitely. Saving and quitting is the fastest way to back this up, but doing so still costs several minutes. This glitch is called "coolgame" and can be seen here.
In Phase 2, when Argorok's back's crystal is vulnerable at the end of a cycle, clawshotting towards the crystal but retracting the clawshot on the same frame it would grab the crystal will result in Argorok being unclawshottable for the remainder of the cycle.