My first destination was the plant pathology department because—well, what a better place to find out about plants?
Is it common/idiomatic to ask a question with that structure?
The idiom you're looking for is what better place, not 'what a better place'. It's commonly used as a comparison with 'than', but that does not preclude it from being used in a case such as yours.
Here's an example:
It should be What better place [is there] to find out...?.
?What a better place is not the same. I don't believe that phrase actually means anything: if it does, it would be similar to What a good place... which is not a question.