What are some examples of self-adjoint functors? Is this an example?

This is a relatively uncommon scenario, but here are a few more examples.

• As you suggest, $(-)^\mathrm{op} : \mathbf{Cat} \to \mathbf{Cat}$ is self-adjoint. More generally, this should be true for the underlying category of any 2-category with a duality involution.
• If $\mathscr C$ has biproducts, so products coincide with coproducts, then the composite $\oplus \circ \Delta_n$ of the (discrete $n$-ary) diagonal functor $\Delta_n$ with the ($n$-ary) biproduct functor $\oplus$ is self-adjoint.
• For completeness, the identity functor is self-adjoint.

There are also similar examples that involve a variance change, i.e. functors $F : \mathscr C^\mathrm{op} \to \mathscr{C}$, such that $F \dashv F^\mathrm{op}$. These are called self-adjoint on the left. The name comes from the characterisation in terms of hom-sets of $\mathscr C$, i.e. we have natural isomorphisms $\mathscr C(F(A), B) \cong \mathscr C(F(B), A)$. Conversely, if $F$ is self-adjoint on the right, then we have natural isomorphisms $\mathscr C(A, F(B)) \cong \mathscr C(B, F(A))$.

• The contravariant powerset functor $\mathcal P: \mathbf{Set} \to \mathbf{Set}^\mathrm{op}$ is left-adjoint to $\mathcal{P}^\mathrm{op} : \mathbf{Set}^\mathrm{op} \to \mathbf{Set}$, i.e. self-adjoint on the right.
• More generally, in a symmetric monoidal closed category $(\mathscr C, \otimes, I, \multimap)$, for a fixed object $A$, the functor $(-) \multimap A$ is self-adjoint on the right.
• In a similar vein, functors self-adjoint on the right are used in Thielecke's Categorical Structure of Continuation Passing Style to describe the structure of CPS (see Example 4.3.2).