What I like:

Here, there is no pushing; you simply line up at the back. It's completely natural.

There are many public and free toilets, for example in subway stations or at 7-Eleven. In subway stations, there are seated and standing toilets, and you can even check if your favorite toilet is available.

I also discovered a free water dispenser at a tourist spot. These are also available in subway stations and other public places, some even with hot water. It might be that you want to brew a cup of tea.