  The recently synthesized air-insensitive hydrogen doped KCr$_3$As$_3$
superconductor has aroused great research interests. This material has, for the
first time in the research area of the quasi-one-dimensional Cr-based
superconductivity (SC), realized a tunability through charge doping, which will
potentially significantly push the development of this area. Here based on the
band structure from first-principle calculations, we construct a six-band
tight-binding (TB) model equipped with multi-orbital Hubbard interactions, and
adopt the random-phase-approximation approach to study the hydrogen-doping
dependence of the pairing symmetry and superconducting $T_c$. Under the
rigid-band approximation, our pairing phase diagram is occupied by the triplet
$p_z$-wave pairing through out the hydrogen-doping regime $x\in (0.4,1)$ in
which SC has been experimentally detected. Remarkably, the $x$-dependence of
$T_c$ shows a peak at the 3D-quasi-1D Lifshitz transition point, although the
total density of state exhibit a dip there. A thorough investigation of the
band structure reveals type-II van-Hove singularities (VHSs) in the $\gamma$
band, which favor the formation of the triplet SC. It turns out that the
$\gamma$- Fermi surface (FS) comprises two flat quasi-1D FS sheets almost
parallel to the $k_z=0$ plane and six almost perpendicular tube-like FS sheets,
and the type-II VHS just lies in the boundary between these two FS parts.
Furthermore, the $\left|k_z\right|$ of the VH planes reaches the maximum near
the Lifshitz-transition point, which pushes the $T_c$ of the $p_z$-wave SC to
the maximum. Our results appeal more experimental access into this intriguing
superconductor.
