I know that by Tennenbaum's Theorem, non-standard models of arithmetic "don't know" which of their elements form a standard model of arithmetic. However, often facts that are opaque to the model from an "internal" perspective can be recognized "from the outside". (After all, if this weren't possible we wouldn't know that these models were non-standard.) For example, countable models of set theory can be seen to be "non-standard", in a sense, because they classify certain sets as uncountable when "from the outside" we can see this is simply a weird side-effect of that model lacking an injective function from these sets into their natural numbers.
So, suppose that from an external perspective we know which substructure of a non-standard model of arithmetic contains the standard numbers (even though the model can't identify this substructure from its perspective) -- perhaps we started with a standard model and constructed the non-standard model from it.
Could the construction of the non-standard model add to the properties of the standard numbers we started with? Or does the construction of the non-standard model always add completely disconnected structure to the initial structure, such that the properties of the initial structure remain completely unchanged?
Since each element of $\mathbb{N}$ is definable, if $M$ is a nonstandard model of true arithmetic then the inclusion map is elementary. So the standard elements satisfy all the same sentences they did originally.
If instead of true arithmetic we are looking at PA (or similar), then even $\Pi^0_1$ facts need not be preserved: e.g. consider the formula $\varphi(x)\equiv$"PA is inconsistent" (note that $x$ is just a dummy variable here). If $M$ is a model of PA + $\neg$Con(PA), then $\varphi(0)$ holds in $M$ but fails in $\mathbb{N}$; and $\varphi$ is $\Pi^0_1$.
Conversely, if $M$ is any $\{+, \times, <\}$-structure with a unique initial segment isomorphic to $\mathbb{N}$, let alone a nonstandard model of PA, then - via that isomorphism - $\mathbb{N}$ embeds in $M$ in a $\Sigma^0_1$-preserving way. So we can never make a true $\Sigma^0_1$ fact become false in a nonstandard model.
A less trivial example of a $\Pi^0_1$ formula whose truth value on a standard number can change from "true" to "false" when we pass to a nonstandard model: let $\varphi(n)$ be "the sentence with Godel number $n$ is consistent with PA." Note that the reason for this formula "fluctuating" is the same as the above, it's just that here we don't have a dummy variable.
In fact, since each standard natural is definable, we are really just talking about sentences changing truth value.