The counterpart you may intend to use is that every cardinal is comparable with $\kappa$. This means that every infinite set which is not smaller than $\kappa$ has a subset of cardinality $\kappa$.
This is the restriction of the well-ordering principle to $\kappa$, whereas $\sf AC_\kappa$ is the restriction of the axiom of choice to families of size at most $\kappa$.
It should be noted that $\sf WO_\kappa$ does not imply $\sf AC_\kappa$, and the other way around is true only if $\sf WO_{<\kappa}$ holds and $\sf AC_{\operatorname{cf}(\kappa)}$ holds for limit cardinals.
For more on that, see Jech The Axiom of Choice, Chapter 8.
Your idea, by the idea is not going to work. It is consistent that we can add an infinite set which have no countably infinite subset, and its rank is arbitrarily high.