Bioshock Infinite Walkthrough Part 32: Guns For The Lady (Finkton: Gunsmith to Handyman Fight)
After going through the double doors you will find the machinery but no one around. Start by going over to the left, around the partition. Turn to the left then and you will find some barrels to loot and Shotgun ammo. Turn around after that and head over to the right. Take the stairs up and you will find a small Buddhist shrine. Elizabeth has a little to say on it, as does Booker. After that continue over to the left. You will find a workspace. Head over to the left and look for the work table with a vice on it. Look on the left-hand side of it to find a lockpick. Just after Booker calls out to Chen, someone will appear downstairs, crying. Head back to the shrine. You will find Mrs. Lin standing there praying for her husband. With all this in mind it is time to head on out.
Once outside head over to the right and go behind the barricade going to the right. If you have not alerted the authorities here, you need to now. Deal with them, but try to save the turrets as they are about to become very useful.
Handyman:
Health: 9952, Ranged: 1334, Melee: 800, Special Attack: Charged Slam, Special Damage: 1382
The Handyman has numerous Melee attacks that he can use. In addition to that he has a charged slam move that not only hurts but will stun Booker if it connects for several seconds. This can easily get him killed on the higher difficulties. He can also launch balls of electricity at Booker given enough time. They can also leap up onto Sky-lines, when present and electrify the whole line. This, of course, means he is immune to Shock Jockey, but they are also immune to Possession and Bucking Bronco.
When Booker emerges outside the Handyman will appear and come after him. These are very hardy foes and not easy to take down. The best thing to immediately do is arm yourself with Murder of Crows and send a flock at the Handyman. As he bats at the Crows, aim and fire at the red heart in the center of his chest. You can also unleash Devil's Kiss on him as well to get some more damage in. Still, the best damage is going to be from unloading the Hand Cannon or another short range, high damage gun into his chest. If you can possess a Turret or a flying Mosquito then you will have some help that can distract the Handyman from Booker. If Booker takes a lot of damage and his shield breaks immediately go for one of the Freight Hooks. Keep moving from hook to hook while the shield recharges to avoid the ranged attacks the Handyman is capable of.
You can also use this advanced mobility against the Handyman. Leap from the ground to a hook then onto one of the awnings. As the Handyman lands near Booker to attack him, hit the Handyman with a Murder of Crows and immediately unload your weapon into his chest. As he recovers you will want to run and grab onto the freight hook. It might take longer but it will help reduce the damage taken.
When the fight is over, be sure to search around the Handyman's corpse. A Handyman will always drop a Gear Box in addition to whatever is found on his corpse. Not too bad for a really hard fight. Keep it in mind for the future. Head over to the theater with the marque talking about Booker DeWitt Auditions. This is going the way you think it is.