Well, in ZFC such a surjection cannot exist, because $|P(A)|>|A|$, and if $|A|\ge\aleph_0$ (that is, $A$ is infinite), then $|P(A)\setminus A| = |P(A)|$ still $>|A|$. (however you mean $P(A)\setminus A$ - probably via the canonical embedding $a\mapsto\{a\}$). Whilst, if there is a surjection $A\to B$, then $|A|\ge|B|$.
On the other hand, there are set theories, e.g. Quine's New Foundation, where Russell's paradox is resolved not by the bigness of the sets, and there there exists the set of everything, say $\Omega$, and there $P(\Omega)$ is contained in $\Omega$. Well, $A=\Omega$ is still not a good choice because $P(A)\setminus A=\emptyset$, but it may make some sense what you ask...