If I were in your position, I'd go for apartment and movie. These are standard American terms, but they're also used worldwide by most other English speakers. Sneakers and subway are less common outside the US, but they're still universally understood. I definitely wouldn't use "tube" - that's an informal name for the London underground.
Here's a fact. If you refer to your home in Japan as an "apartment" or your shoes as "sneakers", it will sound fine to a British person. We wouldn't bat an eyelid. English speakers from Canada, the UK, Ireland, Australia, NZ, SA and the Caribbean are all essentially bilingual when it comes to receptive skills. We have been exposed to so much American English in the media that we're barely even aware of it.
We wouldn't necessarily call our shoes "sneakers" or talk about "going to the movies", but we wouldn't really notice if we met another person who did use these terms. We hear them all the time.
The reverse is not necessarily true. The language environment of most Americans is predominantly American. So a US English speaker who heard you say "flat" would understand you, but they would also probably think, "Huh? Flat? That's kinda weird. Why's this Japanese guy talking like a Brit?"
If in doubt, use the American term - - it's probably your simplest option.