We demonstrate optical cycling and sub-Doppler laser cooling of a cryogenic buffer-gas beam of calcium monohydride (CaH) molecules. We measure vibrational branching ratios for laser cooling transitions for both excited electronic states A and B. We measure further that repeated photon scattering via the A←X transition is achievable at a rate of ∽ 1.6×106 photons/s and demonstrate the interaction-time limited scattering of ∽ 200 photons by repumping the largest vibrational decay channel. We also demonstrate a sub-Doppler cooling technique, namely the magnetically assisted Sisyphus effect, and use it to cool the transverse temperature of a molecular beam of CaH. Using a standing wave of light, we lower the molecular beam's transverse temperature from 12.2(1.2) mK to 5.7(1.1) mK. We compare these results to sub-Doppler forces modeled using optical Bloch equations and Monte Carlo simulations of the molecular beam trajectories. This work establishes a clear pathway for creating a magneto-optical trap (MOT) of CaH molecules. Such a MOT could serve as a starting point for production of ultracold hydrogen gas via dissociation of a trapped CaH cloud.