If something is not false, is it true? For proving vacuous truths for empty sets, Halmos in his Naive Set Theory mentions that


  To prove that something is true about the empty set, prove that it cannot be false.


But now consider Russell's paradox where $A$ is the set of all sets not contained in it. $A \notin A$ being false there doesn't imply that $A\in A$.

So how does Halmos' method work?
I don't know what point you're trying to make about Russell's paradox, but all Halmos is doing is reminding you to try proof by contradiction. Answering your title question with "yes" is equivalent to asserting the law of the excluded middle, which is assumed in classical logic. If you had instead asked, "If something is not true, is it false?", an answer if "yes" would be equivalent to the law of non-contradiction. This law, which Halmos wants you to use, applies even in some non-classical logics (it fails in paraconsistent logics).
Russell's paradox is a pathology that arises because the definition of "set" is too vague in traditional set theory. That is, the problem is with the "definition" of $A$ as "the set of all sets that don't contain themselves" and furthermore assuming that the symbols "$A\in A$" and "$A \not\in A$" can be interpreted with the same semantics as usual for this specific object $A$. 

This is different than the law of the excluded middle, which is the principle in classical logic that $p\vee \lnot p$ is a tautology for all propositions $p$. Hence we have the universally valid inference $(p \vee \lnot p) \wedge \lnot(\lnot p) \Rightarrow p$.

The point that Halmos is making is that due to the law of the excluded middle, you can do proofs by contradiction. For some of the facts about the empty set, this is  very useful. 

Not all systems of logic accept the law of the excluded middle. Non-classical systems like intuitionist or paraconsistent logic do not accept it. In mathematics, there is a school called constructivism that rejects the law of the excluded middle, but most mathematicians accept it.