Most existing computational tools for assumption-based argumentation (ABA) focus on so-called flat frameworks, disregarding the more general case. Here, we study an instantiation-based approach for reasoning in possibly non-flat ABA. For complete-based semantics, an approach of this kind was recently introduced, based on a semantics-preserving translation between ABA and bipolar argumentation frameworks (BAFs). Admissible semantics, however, require us to consider an extension of BAFs which also makes use of premises of arguments (pBAFs). We explore basic properties of pBAFs which we require as a theoretical underpinning for our proposed instantiation-based solver for non-flat ABA under admissible semantics. As our empirical evaluation shows, depending on the ABA instances, the instantiation-based solver is competitive against an ASP-based approach implemented in the style of state-of-the-art solvers for hard argumentation problems.