Is there a proof of the following statement: you cannot prove with natural deduction theorems that are unprovable in a Hilbert-style proof system? The logic in discussion is either propositional logic or FOL.
Yes. You have to follow a "typical" proof of "equivalence" between Natural Deduction and Hilbert-style.
Sara Negri & Jan von Plato, Structural Proof Theory (2001), page 41-on.
An alternative approach is through soundness and completeness.
Both proof systems are sound and complete regarding valid formulae; thus, a formula unprovable in ND, being not valid, in not provable in H-s, and vice versa.