Location: Dressrosa City Area
Petitioner: Barto Club Apprentice
Reward: Barto Club Badge, 74,000 Berries
Objective Reward: Offal Soup with Sea Pig

To unlock this Side Story you need to first free Dressrosa from the Donquixote Family. This means climbing the New Royal Plateau and taking down Doflamingo, the head of the family. Once you've done that and proceeded to Chapter 9, head back to Dressrosa to find a slew of new Side Stories. This particular Side Story is located in the north of Dressrosa City East, at the city's northeastern-most point on the map.

The man is the member of the Barto Club, a group of pirates who are huge fans of the Straw Hats, and he's particularly enamoured with Usopp. He wants Usopp to make him a particularly dangerous Trick Ball, the Perfected Exhaust Ball Plus, as a memento of their meeting. 

There are two ways to get this particular item. The first, more obvious route is to scrounge together the ingredients needed for Usopp to craft the Trick Ball. Usopp needs four ingredients to create a Perfected Exhaust Ball Plus. They likely appear in multiple locations, but I found them specifically in these areas:
  • Mezcal Cactus: Alabasta - Sandy Ruins
  • Negative Seed: Alabasta - Great Sandy Desert, near the Sandy Oasis
  • Unlucky Herb: Alabasta - Sandy Ruins
  • Nimbus Spider: Dressrosa and New Royal Plateau
The chances are excellent that you already have the items you need, but if not, you'll need to check randomized items at the places listed above. Smashing containers throughout One Piece Odyssey will also find you what you want... after a while, anyway. You can assemble the Perfected Exhaust Ball Plus at a campsite or tavern. 

The alternative method is to earn the Perfected Exhaust Ball Plus as a reward for completing various Objectives:
If you've already used up all four of these Trick Balls - unlikely, but who knows - you'll have to make your own. Regardless of the route you take, deliver the Perfected Exhaust Ball Plus to the Barto Club Apprentice to receive your reward.