We investigate the charge and spin transport of a voltage-biased superconducting point contact coupled toa nanomagnet. The magnetization of the nanomagnet is assumed to precess with the Larmor frequencyωLwhen exposed to ferromagnetic resonance conditions. The Larmor precession locally breaks the spin-rotationsymmetry of the quasiparticle scattering and generates spin-polarized Shapiro steps for commensurate Josephsonand Larmor frequencies that lead to magnetization reversal. This interplay between the ac Josephson current andthe magnetization dynamics occurs at voltages|V|=ωL/2enforn=1,2,..., and the subharmonic steps withn>1 are a consequence of multiple Andreev reflection (MAR). Moreover, the spin-precession-assisted MARgenerates quasiparticle scattering amplitudes that, due to interference, lead to current-voltage characteristics ofthe dc charge and spin currents with subharmonic gap structures displaying an even-odd effect.