According to the Second incompleteness theorem $Con(PA)$ is independent of $PA$. So if $PA$ is consistent $PA + \neg Con(PA)$ is also consistent which means that there exist a number $t$ which codes a proof of $1=0$. But $t$ isn't a "regular" natural number because $PA + \neg Con(PA)$ is a nonstandard model of $PA$.
My question is, how do we know that the Gödel's sentence of $PA + \neg Con(PA)$ exists (its code is a "regular" natural number)?
On
The construction of the Gödel for any theory that includes PA depends only on what the axioms of that theory are, textually. It does not depend on having a particular model of the theory, or what the properties of that model is, but proceeds entirely in the metatheory, where we assumes the integers are just the standard ones.
This holds in particular for $\mathrm{PA}+\neg\mathit{Con}(\mathrm{PA})$.
I think there's a bit of confusion here.
First, "$PA+\neg Con(PA)$ is a nonstandard model of $PA$" is false. "$PA+\neg Con(PA)$" is a set of sentences. It can be true, or false, in nonstandard models, but it is not a model itself.
As to how we know the sentence "$\neg Con(PA)$" (which is the only weird sentence in $PA+\neg Con(PA)$) exists: we can actually explicitly write it down! Godel's original paper gives a recipe for how to do this. Actually carrying the recipe out all the way is lengthy, but has been done - see e.g. What does a Godel sentence actually look like?. And there are much more efficient ways to do it, too.
EDIT: Just to be clear, that example Godel sentence is NOT MINE - it was constructed by Hagen von Eitzen.