My girlfriend has dual US - JAP Citizenship.
I am an AUS - NZ dual citizen thinking of applying for UK Citizenship as well.
Will we be able to pass all of these to our children?
If not then can we pass down any combination of 4?
This question has a good answer in the duplicate, but for your specific situation: UK, USA, AUS, and NZ are no problem to hold multiple citizenships. However, it can get expensive to maintain up-to-date passports for all of them. The one you may have more difficulty with is Japan, as Japanese nationality law does not generally support multiple citizenships (but, it's complicated).
Each country decides its own rules for determining who is or can be a citizen, and there is no overarching rule that places a maximum on the number of citizenships that an individual may hold.