How can we conclude that something is unarisen and eternal?

Using our conventional methods of learning we cannot know eternity (or the now). However, you may glimpse this from a higher state of consciousness where the concept of time falls away.

I recall watching eighties music videos a few weeks ago where this falling away of time happened. It was perfectly prestine that the events in the videos happened now. I began to experiment. I explored my past and found that I could intuit my entire life in a single thought form.

This falling away of conceptual time stayed with me for around a week. At the same time I lost my concept of distance and size but those came with entirely different experiences.

Speaking to your question: in essence, you cannot know the answer to your question through the finite constraints of the conditioned mind. Moreover, eternity may be the wrong word. 'nowness' fits better. Everything happened 'now'.

This question along with a number of your other questions lately suggest you are arriving at a crossroads in your understanding. This is a very delicate crossroads and is fraught with danger, because if you choose the wrong fork you could very easily fall into the view of nihilism which is the road leading straight into the avici hells. Therefore, please take care and monitor your thoughts for signs of this and if you discover you are traveling this perilous road please back up and consider that you've made an error due to misunderstanding. Now, with that caveat I'll try to give a meager account of the answer according to my poor understanding...

The answer to your question can be found in chapter 25 of Nagarjuna's MMK. I've posted the first eight verses along with a short and poor paraphrase of the Incomparable Je Lama Tsongkhapa's explanation. I wish you all the best in studying and would suggest to read the real thing in Ocean of Reasoning. I really think a detailed study of this book and in particular the 25th chapter on Nirvana is the best you can do to resolve your question. With that said, here you go:

25.1 If all this is empty. Then there is neither arising nor passing away. Through the abandonment or cessation of what Does one aspire to nirvana?

I've italicized this verse intentionally even though it is not italicized in the original. Why? Because this verse is given from the view of the objector. You can think of Nagarjuna's MMK as a conversation or debate between two interlocutors. The italicized verses are given from the perspective of ignorance or those who object to the correct Middle Way view. The bold verses are given from the perspective of wisdom or those who uphold the correct Middle Way view.

This verse's perspective is one of eternalism. Thinking that rejecting the inherent existence of "all this" is tantamount to saying that Nirvana does not exist and thus falling to nihilism.

25.2 If all this is nonempty, Then there is neither arising nor passing away. Through the abandonment or cessation of what Does one aspire to nirvana?

This verse is the response. It gives the contrary consequence that if one takes the view of eternalism (ie., essentially existing) that Nirvana becomes impossible since all things are fixed and eternal: that is they do not arise or pass away.

25.3 Unrelinquished, unattained, Unannihilated, impermanent, Unarisen, unceased: This is how nirvana is described.

This is the view from the Middle Way and is the correct view and this is the Buddha's own account. This verse is saying that the Middle Way does not maintain that Nirvana is the subsequent elimination of previously essentially existing afflictions which is the view of eternalism. Why? Because this necessarily entails the error that Nirvana would be impossible because from the view of eternalism there is neither arising nor ceasing. How does Nirvana exist or what are the characteristics of Nirvana? It does not exist essentially as desire that has been relinquished, nor essentially as fruits that have been attained, nor as essentially the owning of the aggregates which have been annihilated, nor as the non-empty which would be permanent. It is not essentially arisen nor essentially ceased. It only exists because things do not exist essentially.

25.4 Nirvana is not a thing. It would then have the characteristics of aging and death. No thing exists Without aging and death.

This is a rejection of the extreme of the eternalist. If Nirvana was a thing, then it would have the characteristic of aging and death just as you have pointed out. That is why we can be confident that it is not a thing because the Buddha has described Nirvana as not subject to aging and death.

25.5 If nirvana were a thing, Nirvana would be compounded. A non-compounded thing Does not exist anywhere.

This is a reiteration of the above point from the contrapositive and a rejection of the conception of Nirvana as a thing. It is not a thing because if it were, then we'd have the absurd consequence that Nirvana was compounded and therefore dependent.

25.6 If nirvana were a thing, How could nirvana be nondependent? A nondependent thing Does not exist anywhere.

If Nirvana was a thing, then it would be dependent and this is absurd. For dependent things arise dependent upon causes. All we've seen so far is a rejection of the extreme of eternalism with regard to Nirvana. This is quite in keeping with what your questions are pointing at. Next, we're going to see rejection of Nirvana from the extreme of nihilism. This is what you are in danger of.

25.7 If nirvana were a non-thing, How could it make sense for it not to be a thing? Where nirvana is not a thing, It cannot be a non-thing.

This is the rejection of the view of nihilism saying that Nirvana is a non-thing. What is a non-thing? It is the mere termination of birth and the afflictions. This is the supposition of true cessation. This is a subtle and hard to understand point so please think very carefully about this. Were Nirvana to be a non-thing, then it would be the true cessation of birth and the afflictions. It would be a true ending. And true endings arise only from believing that things exist inherently. It is precisely because inherent existence is impossible that true endings cannot be posited. I think of true endings as inherent existence sneaking in through the back door after eternalism has been refuted. In a sense, eternalism and nihilism are two sides of the same coin of the view of inherent existence.

25.8 If nirvana were a non-thing, How could nirvana be nondependent? Whatever is nondependent Is not a non-thing.

Nirvana is not a non-thing, because if it were then it would have to exist inherently as a non-thing. And inherent existence is an impossible mode of existence. Again, to understand what "non-thing" refers to here, understand that those who assert Nirvana is a non-thing are those who believe that Nirvana is the true exhaustion of the afflictions and the non-existence of rebirth that occurs through the power of actions. This is the view of many Buddhists who believe that with the breakup of the body of an enlightened being that something truly existent has ended. That some mind or being or some thing that was truly there before has now truly and totally disappeared. Something existent has been turned into a non-existent. This is a true ending and it necessarily implies that whatever has been turned into a true non-existent was therefore previously a true existent.

To put this into my own words, I'd say that when you view Nirvana as some thing then the view of it existing in a true, essential, and inherent way rears its head: ie., eternalism.

When you view Nirvana as arising after some thing ends - aka view Nirvana as that non-thing - then the view of that previous thing existing in a true, essential and inherent way rears its head and its true ending pops up: ie., annihilationism.

When you view Nirvana as both some thing that previously existed and subsequently as some non-thing that takes its place then the views of eternalism and annihilationism both rear their head in a mass of confusion that necessarily is caused by believing that inherent existence is a possible mode of existence.

When you view Nirvana as some thing that is "neither a thing nor a non-thing" then again you've misunderstood and are just heaping confusion upon confusion.

The only way to understand Nirvana is to understand emptiness first and then to realize that Nirvana - just like all things - are completely empty of inherent existence. When you understand this directly in an unmistakable and incontrovertible manner then knowledge consequently destroys ignorance. It is just this that is said to be Nirvana.

Hope this helps!

Imagine, for example, that you're trying to travel to Paris (or to Rome, or Varanasi, or anywhere else).

So, you ask people, "How to get to Paris?" -- and you follow road signs, which say, "This way to Paris."

When you get to Paris, after you arrive, I don't think you ask yourself, "How do I know this is Paris?"

I think that descriptions of nirvana (like "unarisen") should be understood as instructions or directions like this.

I don't know, maybe another analogy is like, you go with someone to the sea-shore, and they say, "That ocean is always there." You look, and what you see are waves which appear and disappear, and so it's difficult to see what's constant, to see that there is anything that's constantly there.