Am guessing it may be easy to show that many exact problems (and perhaps strong approximation problems as well) are NP-hard on expanders. The idea is that if you take an arbitrary constant degree graph $G$ on $n$ vertices, and add another expander $H$ on $n$ disjoint vertices, and put a matching between $G$ and $H$, then you get an expander. The reason being that any set of less than half the vertices, will have either a constant fraction of the matching edges outside it, or its intersection with $H$ will have at most say $0.51$ fraction of $H$'s vertices.
Since you can choose $H$ arbitrarily (say take a random graph) you can know the optimal solution for your NP problem in $H$, and hence there may be hope (depending on the problem), that given a solution for the combined graph you can get at least an approximate solution for $G$. But I didn't verify this for any concrete problem.
Of course, as mentioned above, there are natural problems (most notably unique games) where one cannot do such tricks and in particular algorithms are known for expanders and not known in the general case. One should also be able to come up with some contrived example of a problem that's NP hard in general but easy on expanders (e.g., take some arbitrary NP hard problem on graphs, and modify it so that all instances with spectral gap more than $1/\log n$ are YES...).