Yes, "inherent" modifies "problems" in your example. The usual grammatical treatment begins with a relative clause modifying a preceding noun: "the many problems which are inherent in the setup" -> "the many problems inherent in the setup" (by WHIZ, a rule that deletes "which is/are") -> "the inherent problems" (if "in the setup" is not present, the adjective "inherent" is preposed to the noun "setup").
The exact conditions on the preposing of the adjective to the preceding noun are somewhat obscure. The presence of a modifying adverb makes no difference: "the partially inherent problems", but in the given example, we see that a complement to the adjective, "to the setup", does prevent preposing: ?*"the inherent-to-the-setup problems". (Although, as seen here, if you really need to prepose by pretending that "inherent to the setup" is all one word, you can do that.)