The Global Travel Agency in Dobuita is just opposite Tom's Hotdog truck - and in fact, Tom pointed it out back on Disc 1 when you were looking for Chinese people.
Go to the Global Travel Agency first - at ¥200,000 it's a little out of your reach, but you'll get a flier. They don't sell one-way tickets either, so you're not going to be able to afford flights at that price. Head back home - Fuku-san might have more ideas.
If you stop in at the shrine, you'll discover that Megumi's kitten Mimi has gone missing and she asks you to help her find her. Keep walking towards your house, and when you reach the steps on the left-hand side of the road, you'll hear her meowing. Go down the steps, and you'll find Mimi sitting beside one of the houses. Nozomi shows up just as you find the kitten, and the two of you will take her back to Megumi at the shrine.
Back home, talk to Fuku-san some more. He'll decide to help you out, and breaks open his piggy bank to give you his savings. He points out that Nozomi often travels to Canada to visit her parents, and suggests asking her about cheaper transportation than flying.
Nozomi will be back at Aida Florist. Although she's doubtful you'd make it to Hong Kong on the little money you've got, she'll tell you that while she often flies to Vancouver, her grandmother hates planes and takes a boat instead. A one-way ticket on a ferry might be cheaper than flying, so head back to Global Travel.
Predictably, they don't sell boat tickets. Ryo will ask if there are any other travel agencies in the area, and the clerk will reluctantly suggest Asia Travel, pointing out that they have a bad reputation. Regardless, Ryo has to find Lan Di, so it's off to Asia Travel - they're opposite Aida Florist.
Inside, you'll find the most uncooperative, lazy store worker you'll come across in Shenmue. She's too busy reading her magazine to pay you much attention, but when Ryo bugs her enough she'll hand him a boat flyer.
It's well within your budget, and Ryo can't hand over the money quick enough. The clerk isn't so thrilled, and tells you that the ticket will be ready in three or four hours.
You've got more time to kill - I suggest heading to the parking lot and getting a workout in before you go back. You'll not have a whole lot of money on-hand to spend in the Arcade, and stronger moves are extremely beneficial pretty soon.
Once you've passed enough time, go back to Asia Travel. The clerk from before has disappeared, and instead you'll find a couple of guys hanging round looking shady. The guy behind the desk claims he doesn't know anything about any woman who works there, and refuses to give you a ticket.
After beating up the guy's friends, Ryo will persuade the clerk to get him another ticket and you'll be told to wait for a phone call tomorrow.
In the morning, you'll get a call from Asia Travel - the guy's name is Jimmy - telling you he'll have your ticket at the Arcade at noon. Ryo curtly tells Fuku-san he's going out, and then you're on your way back to Dobuita, for another meeting at the You GAME Arcade.
"Noon" isn't as specific a time as it might sound; in fact, you can enter You GAME whenever you like. Go into the Arcade and the skinhead, Chai, shows up - he'll eat your ticket, then take you on in a Free Battle.
You can win, but it's very difficult. You'll need to have your moves well trained up, focus on low attacks (kicks like Mud Spider and Crawl Cyclone work well), and dodge lots. Throws aren't much use; Chai will block most of them, and he's too fast to get close enough most of the time.
If you do manage to win you'll chase Chai to Asia Travel, which starts the QTE below. If you get beaten, Fuku-san shows up - Chai cuts and runs, and Fuku-san will carry Ryo home to recuperate overnight. In the morning, you can head back to Asia Travel to chase Jimmy through Dobuita, one of the longest QTEs in the game.
You'll catch Jimmy by the Arcade, and drag him back to Asia Travel to ask him some questions - on disc 3.