Start by exploring the area as usual for freebies before heading to Chen Lin's Gun Shop. There is a Gear to be picked on the gun shop's sign board too. Just use the freight hooks to gain height and reach it. Now in the "lobby" before entering the actual Gunsmith Shop, you'll find a Voxophone on the counter. Next, you'll want to head to plaza's "dock" of sorts and pick the lock of the door there. Inside is another coded message that requires a Code Book to decipher.
The Code Book is in the Good Time Club so you can't do anything about the puzzle yet. but do keep this spot in mind for later. Explore the rest of the plaza while keeping a look out in the air for a line floating cargo, follow that to a stockpile of logs. Leaning on the table saw here is a Voxophone.
Enter the Gunsmith Shop now. Go all the way upstairs until Booker finds the place empty, then head back down to altar and speak to Mrs. Lin. You will walk right into a Handyman ambush as soon as you leave the Gun Shop. Possess the gun turret to help you in the fight or destroy it.
Either way, it's up to you. The Handyman is a brute and attacks aggressively. Vigors have little to no effect so it's best to stick with guns. Kill the Handyman by exclusively shooting his heart earns you the Heartbreaker achievement. Pick up the Gear the Handyman left behind before leaving.