  Let $\tau_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator in a
bounded domain $\Omega$ of the form $\Omega_{\text{out}} \setminus
\overline{B_{\alpha}}$ under the Neumann boundary condition on $\partial
\Omega_{\text{out}}$ and the Robin boundary condition with parameter $h \in
(-\infty,+\infty]$ on the sphere $\partial B_\alpha$ of radius $\alpha>0$
centered at the origin, the limiting case $h=+\infty$ being understood as the
Dirichlet boundary condition on $\partial B_\alpha$. In the case $h>0$, it is
known that the first eigenvalue $\tau_1(\Omega)$ does not exceed
$\tau_1(B_\beta \setminus \overline{B_\alpha})$, where $\beta>0$ is chosen such
that $|\Omega| = |B_\beta \setminus \overline{B_\alpha}|$, which can be
regarded as a reverse Faber-Krahn type inequality. We establish this result for
any $h \in (-\infty,+\infty]$. Moreover, we provide related estimates for
higher eigenvalues under additional geometric assumptions on $\Omega$, which
can be seen as Szeg\H{o}-Weinberger type inequalities. A few counterexamples to
the obtained inequalities for domains violating imposed geometric assumptions
are given. As auxiliary information, we investigate shapes of eigenfunctions
associated with several eigenvalues $\tau_{i}(B_\beta \setminus
\overline{B_\alpha})$ and show that they are nonradial at least for all
positive and all sufficiently negative $h$ when $i \in \{2,\ldots,N+2\}$. At
the same time, we give numerical evidence that, in the planar case $N=2$,
already second eigenfunctions can be radial for some $h<0$. The latter fact
provides a simple counterexample to the Payne nodal line conjecture in the case
of the mixed boundary conditions.
