Why is the hitting set problem in NP
You cannot check, in general, whether $S'$ contains at least one element from each subset $C$ in polynomial time w.r.t. $n$.

However, in order to show that the problem is NP it suffices to show that you can check to above condition in polynomial time w.r.t. the instance size, which is in $\Omega( \sum_{X \in C} |X| + \log n)$.