Given that according to Gödel's theorems there are propositions in any language equivalent to first order logic that cannot be proven right or wrong, is it possible to prove that some of such propositions are not provable? I am thinking of something like a proof of impossibility applied to propositions.
Gödel showed how to construct a statement in any sufficiently powerful formal system that states its own unprovability within the language of the formal system itself (he did this by showing that propositions in the language of the formal system can be translated into numbers and that "provability" can be translated into artithmetical properties of those numbers). But his construction method creates a specific proposition - it cannot be used to determine the provability of a general proposition.
This is analagous to how Liouville showed that transcendental numbers exist by constructing a family of numbers with properties which allowed him to prove they were not algebraic. It is much more difficult to prove that a particular number such as $e$ or $\pi$ is transcendental.