Let's denote by ZF the Zermelo-Fraenkel axioms, and by AC the Axiom of Choice. I am sorry that the question is not very well formulated, but still:
can there be a "number theoretic statement" (by which I understand a 1st order logic sentence in the language of sets which is true iff a certain "number theoretic" fact is true), such that it follows from ZF+AC, and whose negation follows from ZF+$\neg$AC?
There are no first-order sentence like that.
If $\varphi$ is such sentence, then it holds in $V$ if and only if it holds in $L$, which satisfies choice. So if $\varphi$ follows from choice, it must holds in $L$ and thus in $V$, so it cannot be disproved by $\lnot\sf AC$. And similarly, if $V$ satisfies $\lnot\sf AC$, and $\varphi$ holds there, it must hold in $L$.
In fact, you can push $\varphi$ to a relatively simple second-order sentence, $\Sigma^1_2$ specifically, by Shoenfield's absoluteness theorem.