Fighting Gohdan is relatively straightforward — hit each of his hands with two arrows, followed by each of his eyes, and then throw a bomb into his mouth — amounting to a minimum of 3 cycles. You only need 24 arrows if every shot you take is a hit. It is however possible (although rare) to get double hits.
When attempting to disable Gohdan’s hands, it is important to time your shots accordingly — firing arrows at them rapidly will only deplete your quiver and waste time. Instead, wait a second after getting the first shot on each hand. This will increase the chance that each arrow fired is a hit, as each hand undergoes a brief period of invincibility towards further damage immediately after they are shot.
Sometimes, for an unknown reason, after you have stunned the first hand and are stunning the second, the first hand will randomly respawn. It seems as though this possibility may be lowered by alternating between each hand.
Note: It is possible to hit both hands with one arrow if they line up properly, as well as it is possible to deal damage to the hands with bombs. However due to their relatively random movement, this would only likely be exploited in a TAS.
Once Gohdan’s hands are temporarily stunned, you will have to deal with his eyes. Again, it is key to wait for a second after getting a successful hit on an eye before taking another shot to minimize arrow use. Typically after disabling the first of the eyes, Gohdan will attempt to shoot fireballs at you. This can be avoided by running directly underneath him. However if you do this part of the fight very quickly, there is a possibility to hit his eyes all four times before he charges up. This is very difficult (as well as a bit luck-based) due to his random movement. If you succeed though, it would mean that he never fires at you during the fight — saving some time. After both eyes are disabled, you can simply throw a bomb into his mouth and then repeat the whole process.
If you lose all of your arrows during the fight, Gohdan will give you a drop of 10 new ones an unlimited number of times. Although, if you are losing all of your arrows when the hands are stunned, they will respawn after the arrows have been dropped. Additionally, if you have only one arrow left and you try to use it on Gohdan, not only will it not deal any damage, but you will have to sit through the cutscene of Gohdan providing an arrow drop — ultimately meaning that it is crucial to be exact in your shots as to minimize time loss.
It is possible although extremely difficult to defeat Gohdan without the bow. This is because bombs actually can damage both Gohdan's hands and eyes. With perfectly thrown bombs, you can dispatch of Gohdan without the bow. Note however that if you don't own a quiver, Gohdan will perpetually be dropping green rupees, making the fight even longer and harder. As such, it is advised to obtain a quiver early before attempting this fight without the bow.