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The bullet's past section is essentially a recreation of the Agahnim and Gannon fights in Zelda A Link to the Past. You have nothing but a sword and two blanks, and this sword does NOT destroy bullets.
To start with, realize that having armor going into the past will carry over, even though your health pool doesn't, so try to buy up armor from the shop before you head in if there's any left to buy. You will likely be glad you did.
The first phase is just two of the chain swinging gun nut enemies. Wait until they finish their attack, then move in for 2-3 hits. Don't stay in too long. While you're backed off, watch intently for them to throw it at you. Do not use any blanks for this part.
The second phase requires you to just dodge attacks until the boss shoots a purple ball at you. You have to then knock it back at him by swinging and hitting it. On each subsequent hit, he knocks it back more times, and you need to have quick reflexes. Don't be standing too close or you'll have no time to react. Some of the bullet patterns here might warrant a blank, but try to just dodge or dodge roll if you can.
The third phase is a very long fight because you have to wait for a chance to move in. The #1 rule here is patience. You can not die from taking too long. Wait until you get a good opening, and wait as long as it takes. If he's standing in the bottom corner, you might want to just dodge attacks until he moves again because most of his attacks originate from his top. It'll be nearly impossible to not be hit if you move in from the top. This might also be a good time to use your blanks if you want to move in to get him to move out from the bottom. I found that the best attack to hit him during was when he spins the bullets around in a circle. You can walk up to his bottom and get 3 hits in before you dodge roll out, then get two more when the bullets retract again. If you find yourself about to die, start just dodging everything and ONLY attacking during that ability.