“It came right as” implies that one action (it ) had just occurred at the same time another event is still taking place. You can now describe this vocabulary in an abstract, but lets summarize some of the examples.

If I say “The apple fell from the tree right when I was coming outside” this means that at the same point in time that I was coming out, the apple had just fallen from the tree.

If I said I had finished my homework right when the dog started barking, it means that at the same point in time that I had just finished my homework, the dog then started barking.

In your example, the first action that was just completed is unspecified, probably given in an exterior context. The ongoing action is that “President Trump and the Japanese Minister were sitting down for dinner at the presiden’s resort”.