  In the evaluation of the half-life of the neutrinoless double-$\beta$ decay
($0\nu\beta\beta$) of a doubly closed-subshell nucleus $^{96}$Zr, the structure
of the nucleus $^{96}$Mo is essentially important. The $\alpha$-clustering
aspects of $^{96}$Mo are investigated for the first time. By studying the
nuclear rainbows in $\alpha$ scattering from $^{92}$Zr at high energies and the
characteristic structure of the excitation functions at the extreme backward
angle at the low-energy region, the interaction potential between the $\alpha$
particle and the $^{92}$Zr nucleus is determined well in the double folding
model. The validity of the double folding model was reinforced by studying
$\alpha$ scattering from neighboring nuclei $^{90}$Zr, $^{91}$Zr, and
$^{94}$Zr. The double-folding-model calculations reproduced well all the
observed angular distributions over a wide range of incident energies and the
characteristic excitation functions. By using the obtained potential the
$\alpha$ +$^{92}$Zr cluster structure of $^{96}$Mo is investigated in the
spirit of a unified description of scattering and structure. The existence of
the second-higher nodal band states with the $\alpha$+ $^{92}$Zr cluster
structure, in which two more nodes are excited in the relative motion compared
with the ground band, is demonstrated. The calculation reproduces well the
ground-band states of $^{96}$Mo in agreement with experiment. The experimental
$B(E2)$ value of the transition in the ground band is also reproduced well. The
effect of $\alpha$ clustering in $^{96}$Mo on the the half-life of the
$0\nu\beta\beta$ double-$\beta$ decay of $^{96}$Zr is discussed.
