We present results from MUSE spatially resolved spectroscopy of 21 post-starburst galaxies in the centers of eight clusters from z similar to 0.3 to z similar to 0.4. We measure spatially resolved star formation histories (SFHs), the time since quenching (t ( Q )), and the fraction of stellar mass assembled in the past 1.5 Gyr (mu (1.5)). The SFHs display a clear enhancement of star formation prior to quenching for 16 out of 21 objects, with at least 10% (and up to >50%) of the stellar mass being assembled in the past 1.5 Gyr and t ( Q ) ranging from less than 100 to similar to 800 Myr. By mapping t ( Q ) and mu (1.5), we analyze the quenching patterns of the galaxies. Most galaxies in our sample have quenched their star formation from the outside in or show a side-to-side/irregular pattern, both consistent with quenching by ram pressure stripping. Only three objects show an inside-out quenching pattern, all of which are at the high-mass end of our sample. At least two of them currently host an active galactic nucleus. In two post-starbursts, we identify tails of ionized gas indicating that these objects had their gas stripped by ram pressure very recently. Post-starburst features are also found in the stripped regions of galaxies undergoing ram pressure stripping in the same clusters, confirming the link between these classes of objects. Our results point to ram pressure stripping as the main driver of fast quenching in these environments, with active galactic nuclei playing a role at high stellar masses.