It’s good manners to go around and introduce yourself to immediate neighbors shortly after moving in (even if you don't speak Japanese so well). If you’re in a large apartment building , you can limit your introduction to the neighbors next door, above and below.
You can use the phrases: Hajimemashite! (House number) ni hikkoshite iru (your surname) to mou shimasu. Yoroshiku onegai shimasu. Loosely translation this means, Nice to meet you! I’m (name) from (Apartment number). Let’s cooperate as good neighbors.
It’s recommended that you give your neighbors a small gift — dishtowels, sweets, snacks, or something from your home country.