$NFU$ doesn't prove Cantor's theorem (that $\mathcal{P}(S)$ is cardinally greater than $S$) by a stratification dodge: the proof's critical step makes use of the unstratified formula $x \not\in f(x)$, so does not go through in $NFU$. In particular e.g. the universal set $V$ is as big as its powerset. However, something like Cantor's proof goes through to show that there is no bijection between e.g. $V$ and singletons of elements of $V$, since $\{x\} \not\in f(x)$ is stratified, so a version of Cantor's proof modified in obvious ways goes through.
$NFU$ also doesn't prove the axiom of infinity, by Goedel's second incompleteness theorem (it has a model in the natural numbers, as Jensen showed (!)). My question is, if we add the axiom of infinity ($\omega$ exists), do we know that $\mathcal{P}(\omega)$ is bigger than $\omega$? It seems to me that, without extra axioms (other than stratified comprehension, extensionality, infinity), we won't get this: we won't know anything more than that there are fewer singletons of natural numbers than there are natural numbers, by the same argument as that hinted toward above. Is this right? Seems likely wrong.