Dual US Japanese citizen or Dual US Japan citizen?

I don't know that grammar enters into it, but usage i would expect to see from a native speaker of English would be Korean or Japanese. Japan and Korea are nations, not people.

I think you would say someone has dual US/Japanese citizenship.

This is correct. Citizenship can take an adjective of nationality, like other nouns.

I have a German car.
You took a Spanish holiday.
Sven has Swedish citizenship.

It sounds like you are correct.
That 'Citizenship' takes adjective of nationality.

So does the same rule applies for 'citizen'?
I am little confused since 'US' seems to be noun instead of adjective. So that is why I thought 'Japan citizen' is possible.

For the US, the name (noun) is the United States of America or "the USA" or "the US". The adjective is "United States" or "US", or sometimes "American".

A person can be:

A citizen of (noun)
A or an (adjective) citizen

A citizen of Japan (noun)
A Japanese (adjective) citizen

A citizen of Egypt
An Egyptian citizen

A citizen of the US (noun)
A US (adjective) citizen