How to check if a Nintendo Switch game card supports multiple languages

Out of the box, no. The Nintendo 64 has a region lockout chip which prevents NTSC (Japanese and US) games from running on a PAL (European) machine.

However, through the use of third-party devices, most games should work. I personally own a N64 Passport Plus which I use to play Hey You, Pikachu! on my own N64. It basically works by using a second, local cartridge to authenticate with the lockout mechanism.

I don't have first hand experience of this, but according to that Wikipedia article, some games won't work even with this - presumably, they perform additional hardware detection, so they might be hard to do anything about. It may be possible to somehow bypass these through Action Replay codes; I don't know exactly how these games perform those checks.

You could also buy a Japanese N64 and bring that home. You'll need a power converter to make it run on 230V, and a TV which will accept the input, but you could run any Japanese game that way, and any US game if you remove the plastic tabs.

As an alternative, if you have a European Wii, at least some of the games might be available on the Virtual Console, saving you the trouble of messing with adapters.

It seems like it is more of a technical side issue, furthermore it feels like you already have the answer. That you just need to set it up through your ISP. Most aftermarket technical support company's should be able to walk you through the process. You may need to have two routers running off of the same modem. One with your local ip and one with your Japanese ip, and then just switch back and forth.