Let $A$ consist of the rationals together with $\sqrt{2}$. Then $A$ is countable, so there is a bijection $\phi$ from $\mathbb{N}$ to $A$. For any pair of natural numbers $a$ and $b$, we say that $a\prec b$ iff $\phi(a)\lt \phi(b)$ in the ordinary ordering of $A$.
Then $\prec$ is a "non-standard" ordering of $\mathbb{N}$.
Let $K\subset A$ consist of all numbers $x$ in $A$ such that $x\lt \sqrt{2}$, and let $S=\phi^{-1}(K)$.
Remark: As long as we do not ask for any kind of interaction between the ordering and the ordinary structure of $\mathbb{N}$, anything that can happen in a countably infinite set can be replicated in $\mathbb{N}$.