What I understand so far:
If $S$ is any set then AC gives us a choice function, that is, $f: P(S)\setminus \{\varnothing \} \to S$ such that $f$ returns an element of $A \in P(S) \setminus \{\varnothing \}$.
Assume we have a bijection $f: S \to \mathbb N$. Then $S$ is well-ordered (since $\mathbb N$ is) and we know that we can explicitly give a choice function if $S$ is well-ordered: define $f: P(S) \setminus \{ \varnothing \} \to S$ to be the function that returns the least element of $A$.
Hope my understanding so far is correct.
Now assume we have a set $S$ for which we also assume that we have a choice function $f: P(S)\setminus \{ \varnothing \} \to S$. Does one need AC to assume the existence of such a choice function?