So, I did a bit of Googling, and confirmed that my idea of what 'upstate' meant wasn't too far off. The first entry I found with a search for 'upstate definition', for example, was "of, in, or to a part of a state remote from its large cities, especially the northern part." I also came across more references to upstate New York in particular, which appear to confirm that my (also British) friend in Rochester, NY, counts as living in upstate New York.
My question to US speakers of English is this:
Is the phrase 'upstate + [state name]' often (or ever?) used for states which don't share their name with a major American city?
[Edited for typo]