Are there some fill-in-the-blank type questions that, if one were to fill in the blank and it sounded right to a fluent speaker then it would have to be a proper noun (or it would have to be a common noun)?
It's rare, though not impossible, to have an article in front of a proper noun, e.g. "The Alan...", "A Steve...". But these are not always completely bad: "The Alan with the white shirt...", etc.
Still, even in the latter example I think it sounds awkward enough to provide a decent heuristic, if that's what you are looking for.
I am not sure you can devise such a test, taking into account only two of the many types of nouns (abstract, concrete, non-countable or mass, countable, collective).
The only test I can think of is related to the plural form: Proper nouns rarely have a plural form (at least, comparatively to common nouns).
So, if you can say: "I have/know many xxxxS", and that sounds right to a fluent speaker, chances are: xxxx is not a proper noun.
But that doesn't imply it is a common noun.
The only way I can think of doing this, is by using names. For example:
- Alice visited
__________. Together, they had a nice time.
__________is very tall. He is over six feet.
You could add some examples where the noun is referrer to by "it" to make the distinction. The problem, of course, is that referring to countries or other proper nouns (for instance a company name) isn't possible.