  To study the possible mechanism of superconductivity in Na$_{0.35}$CoO$_2$,
we examine the interaction between all the relevant optical phonons (breathing
and shear phonons) and $t_{2g}(a_{1g}+e_g')$-electrons of Co-ions, and study
the transition temperature for a s-wave superconductivity. The obtained $T_{\rm
c}$ is very low when the $e_g'$-valence-bands are far below the Fermi level.
However, $T_{\rm c}$ is strongly enhanced when the top of the
$e_g'$-valence-bands is close to the Fermi level (say -50meV), thanks to
interband hopping of Cooper pairs caused by shear phonons. This ``valence-band
Suhl-Kondo mechanism'' due to shear phonons is significant to understand the
superconductivity in Na$_{0.35}$CoO$_2$. By the same mechanism, the kink
structure of the band-dispersion observed by ARPES, which indicates the strong
mass-enhancement ($m^\ast/m\sim3$) due to optical phonons, is also explained.
