Sure you can say “on Tuesday”, but I’d argue that that’s an adjectival noun phrase. It’s absolutely acting as an adverb.
I made a valiant argument for the “verb” angle while acknowledging the noun interpretation, but I concede that point. “Rise” is most appropriately regarded as a noun, and “sun” as an attributive (adjectival) noun.
Do you mean “that’s an adverbial noun phrase”? Otherwise I’m really confused.
I don’t know what prompt you used to get that bullshit, but I do acknowledge the point. I no longer have the original prompts I used but obviously the very first question in the OP wasn’t one of them, or if it was, I got drastically different answers. When I tried a cut-and-paste of that exact first question from the OP, GPT assured me that in “sunrise”, “rise” was indeed an adverb. 
I guess I just get frustrated with these kinds of condemnations because the vast majority of the time – though I haven’t been using it much lately – GPT gives me good information confirmed from other sources, and when it doesn’t, the errors are so blatant that they’re obvious.
ISTM that when an AI does something stupid, it gets all over the media and the internet. All the times that it’s amazingly successful are just quietly tolerated.
Bah! Thank you. It must be my bedtime. 
Okay, but this time, the errors were so blatant that they were obvious. If we can’t admit when AI does something stupid, we’re going to have a hard time evaluating it. Freaking out about discussing its occasional foolishness is unhelpful.
If a human writes something that seems obviously wrong, I might spend time figuring out why it’s wrong, for any number of useful reasons.
If a robot writes something obviously wrong, I’m going to say “this robot is an idiot” and move on with my life.
So yes, being able to tell the difference between the two is deeply important, as is understanding where LLMs are useful for professional practice and where the lines are between useful, suspect, and absolute garbage.
I’m real late to the party.
I suppose that if English had settled on the term “sunrising” as a near-synonym for dawn, rather than “sunrise”, the verby nature of the “rising” part would be obvious. e.g.
At the sunrising I was outside to enjoy the air and the sound of awakening birds.
“Sunrising” isn’t a currently idiomatic term, but there are some few other noun/verb combos in ordinary English that are constructed that way. ISTM they’re becoming archaic. My example sentence has an 1800s feel to it. At least to very much non-expert me.
I’d say noun + verb
sun + (to) rise
sun + (to) set
Looking up the etymology for upset, it seems to be preposition + verb
From Middle English upset (“the act of setting up; establishment”), from Middle English upsetten , corresponding to up- + set . Cognate with Middle Low German upset (“setup; arrangement”).
I am surprised that I made it here before the learned Johanna to discuss the linguistics of nominal compounds! ISTM that what we’ve got here is mostly an analysis of different types of determinative compounds:
AFAICT, the concept of a nominal compound combining a noun with an adverb is not really very helpful at all in this context.
Slight tangent on this, for people who remember Quora as being really good back in the day. It has changed drastically in recent years and calling it a garbage site is almost too kind.
They’ve done away with moderation, they actively use AI to answer questions (it isn’t just rogue users doing it, it is Quora itself), and they’ve prioritized spam click-bait questions with BS answers.
This article goes into detail:
Completely agree. The fact that “sun” is attributive is indicated by its bearing the stress in the compund. I cannot think offhand of any noun+adverb compound.
Isn’t “sunrise” just short for “the sun’s rise”? “Rise”, in this case, is a noun.
How about “lightning strike”? I don’t think anyone would argue that the “strike” here isn’t a noun. What’s the difference?