The expression 'any day now' can also be used in a bit of a sarcastic way when you are waiting on someone and want them to hurry up. Example:
(Amy still can't decide what to wear, so Bob is worried they'll be late to the party)
B: Any day now, Amy! (using an impatient tone of voice)
A: Sorry, sorry. I'm almost done!
In my experience, this use of 'any day now' is much more common than the one you have given (I got the quiz wrong, even though I am a native speaker lol), though it is still correct. To me, using 'any day now' to mean 'within the next few days' seems too formal for casual use, and even maybe a little antiquated. For example, if I were at work emailing a coworker or a client, I could say 'it will be ready any day now' (though I think older people are more likely to use this term). But if I were with friends I would say 'it'll be ready soon' or 'it'll be ready in the next few days'.
This is my experience using American English. Perhaps other English dialects are different.