Enter the plaza, turn right and proceed towards the watchmaker's store. In order to get to the store you must stay close to the right side of the plaza, walking under the arcades until you get to the dock. You'll notice crates in front of the locked store and you'll find beer, money and 2 lockpicks here.
Get inside the store which shouldn't be difficult since you only need one lockpick to unlock the door. Notice Vox code on a wall with large clocks. Examining this code will trigger a new side mission (it's about finding the cipher) and you'll have a chance to complete it after advancing to the next chapters.
The area to your right holds another lockpick found on a shelf between clocks. Search the rest of the room and then get back to the plaza.
Explore the area behind Fink's factory (to the left of the job fair) to find a tailor workshop with a very unusual red tear. There's a new lockpick on one of the tables and you'll also find a machine used to upgrade vigors in a nearby smaller room.
Use the sky-hook to get to a small roof located opposite The Good Time Club. You'll find a new lockpick under the "Killing Time Kills Columbia" billboard, along with some ammunition for the grenade launcher.
Once you've explored the roofs proceed to a dead end alley located behind the (locked) entrance to Shantytown. Approach a circular saw to find Voxophone #38.
Climb on the roof of the Chen Lin's store. You shouldn't have any problems finding two items under the store's sign - a sniper rifle and Gear #16 - Torso. Once you've collected these things check the nearby roofs. The most valuable find is a Volley Gun found opposite the gate leading to Shantytown and it'll come in handy after the visit in Chen Lin's store.
You may finally focus on your current objective by entering the store. Explore the entire shop. There's a lockpick and Voxophone #39 on the counter. Once you have these items use the red door to enter the gunsmith's workshop.
Search the ground level after entering the workshop and once you're done get to the last floor. You won't find Chen Lin, but there's a new lockpick on the table in the corner.
Once you've properly searched the top floor of the workshop you should hear someone crying behind the Budda's altar. This person is going to turn out to be Lady Lin and she'll inform you that the gunsmith has been taken to The Good Time Club. This is a good moment to go there.
Expect a surprise attack from the Handyman after leaving the store and sadly this time you won't be allowed to avoid the duel with this person. Running away to nearby rooftops won't do you any good, because the Handyman can traverse the environment at least as good as the main character. He's also very strong and even attacks based on vigor won't inflict serious wounds. The best method to win is to maintain safe distance from the Handyman and either start aiming for his heart or use explosive weapons (grenade launcher, volley gun etc.).
Once the Handyman has fallen you'll have a chance to collect a proper reward for your struggle in the duel - Gear #17 (random type). Finish exploring the plaza before going to The Good Time Club. If you like long distance fights make sure to take a sniper rifle with you (there's one on the rooftop of the gunsmith's store).