It would still be understood but grammatically it has to be "người + bạn".
Based on what you said, that means the given answer is grammatically wrong right?
The given answer is still correct. "một người bạn Phần Lan" is literally as "a Finnish friend" while "một bạn người Phần Lan" can be interpreted as "a friend who is Finnish". Not much difference.
Oops I've edited it, I meant "người bạn (insert nationality)". Người bạn can be interpreted as "friend" so simply adding a tag onto the end of it will specify what type of friend or where they're from. For example "người bạn Pháp" = French friend, or "người bạn tốt" = a good friend. Using the inverted structure would result in a subtly different emphasis (on the actual descriptor of your friend).
This actually is slow for me. But I go to Crown Five before moving on, and have much listening practice.
I've had sentences where I had to listen to them with my eyes closed for a dozen or more times. Takes awhile to rewire my melon.