Are there true arithmetical statements that corresponds to inconsistency of inconsistent theories? Lets take Naive set theory "NvST" which is the theory whose axioms are all instances of naive unrestricted comprehension, which is of course known to be inconsistent. So $\neg$ Con(NvST) is a TRUE statement!

Question is there a TRUE arithmetical statement that corresponds to $\neg$ Con(NvST)?

The problem I'm getting is that since NvST is inconsistent, then it cannot be interpreted in PA + $\omega$-rule. So how can we get a sentence that can speak about the consistency status of that theory in the language of arithmetic if the theory itself cannot be interpreted in that language?
 A: Gödel's work shows us how to write down an arithmetical statement that corresponds to $\operatorname{Con}(T)$ or $\neg\operatorname{Con}(T)$ for any theory $T$, as long as the set of axioms of $T$ is Turing-recognizable.
This works purely syntactically, and does not depend in any way of having an interpretation of the language of $T$ in mind. So you can certainly apply it to your naive set theory, since it is easy to recognize instances of unrestricted comprehension.
Since it is certainly the case that NvST is inconsistent -- you can prove a contradiction in just a handful of lines! -- the arithmetical statement $\neg\operatorname{Con}({\sf NvST})$ is certainly true in $\mathbb N$.

Where interpretability in $T$ comes into play is if we need something like $\operatorname{Con}(T)$ to be a $T$-statement rather than an arithmetical statement. This need arises on the way to the incompleteness theorem. And even then, what matters is that we can interpret a certain amount of arithmetic in $T$, not that we can interpret $T$ in the metatheory.
But we don't need to go there if all we want is to speak about provability in $T$ with arithmetical statements.
