After you use it, you will get a password. Take this password to TamaTown and go to the Travel Center. Then, you'll get your souvenir password. I really think you should be able to actually play in Japan, or wherever your ticket takes you. Or can you in V4 TamaTown? I've only been to the V3 TamaTown.
Make sure your tamagotchi is an adult. If he/she is then go to your items list. Press button A (button on far left) until you have found the ticket. To use it press B (middle button). If it makes that beep noise like it does when it looses a game, then your tamagotchi is not an adult.
If it is an adult and it's doing that, sorry I can't help.