I want to know when a cardinality $\kappa$ admits a (countably additive) probability measure on its powerset that gives probability zero to subsets of cardinality strictly less than $\kappa$.
I am aware of some results in the vicinity. We can strengthen the last condition by requiring that the measure also be $\kappa$-additive. In that case $\kappa$ is real-valued-measurable and various textbooks lay out interesting restrictions on what those can be like. Or we can weaken the last condition to the measure's giving probability zero to all singletons. In that case my question would be easily answered: $\kappa$ has that property if and only if it exceeds the least such cardinal with that property, $\lambda$, because if $\mu$ is a measure on $\lambda$, then we can define a measure $\nu$ on $\kappa>\lambda$ by setting $\nu(X)=\mu(X\cap\lambda)$. I believe Ulam showed that the least cardinal with this property is the least real-valued measurable cardinal.
So what happens when we require that the probability measure give probability zero to subsets smaller than $\kappa$, but allow that it is not $\kappa$-additive? Is it easy to reduce the problem to either of the above two cases?
If you want your measure to only be finitely additive, then this is a theorem of $\sf ZFC$. For any infinite cardinal $\kappa$, consider the filter of co-small subsets (i.e. $\{A\subseteq\kappa\mid |\kappa\setminus A|<\kappa\}$), extend it to an ultrafilter, and that is a finitely additive $0$-$1$ measure.
If you want more substance, and that the measure is $\sigma$-additive, then for the least such $\kappa$, it will necessarily be $\kappa$-additive again, and we're back with real-valued measurable. You can find this in Jech's "Set Theory" (3rd Millennium edition) as Lemma 10.5(i).