You dispose of it where you bought it, or you don't open it until you get home, or you act like a good hiker who is out in the wilderness, and pack up your trash to bring home with you where you can properly dispose of it.
I always found it amazing that Japanese cities manage to stay so clean without public trash cans everywhere. It's a reminder that you have to solve the social and cultural problems first: if people think it's ok to throw trash on the ground, it doesn't matter how many public trash cans you have.
It's also useful to remember that the general lack of public trash bins isn't a long-standing part of Japanese culture: it only dates to the 1990s, when public trash bins were used for the infamous Sarin gas attack. After that, most bins in cities were removed. What's cultural is what happened later: the cities didn't turn into trash heaps, because people simply took their trash home or otherwise waited to find a suitable place to dispose of it.
I always found it amazing that Japanese cities manage to stay so clean without public trash cans everywhere. It's a reminder that you have to solve the social and cultural problems first: if people think it's ok to throw trash on the ground, it doesn't matter how many public trash cans you have.