We present a Bayesian full-spectral-fitting analysis of 75 massive (M_*_>10^10.3^M_{sun}_) UVJ-selected galaxies at redshifts of 1.0<z<1.3, combining extremely deep rest-frame ultraviolet spectroscopy from VANDELS with multiwavelength photometry. By the use of a sophisticated physical plus systematic uncertainties model, constructed within the BAGPIPES code, we place strong constraints on the star-formation histories (SFHs) of individual objects. We first constrain the stellar mass versus stellar age relationship, finding a steep trend towards earlier average formation time with increasing stellar mass (downsizing) of 1.48^+0.34^_-0.39_Gyr per decade in mass, although this shows signs of flattening at M_*_>10^11^M_{sun}_. We show that this is consistent with other spectroscopic studies from 0<z<2. This relationship places strong constraints on the AGN-feedback models used in cosmological simulations. We demonstrate that, although the relationships predicted by SIMBA and ILLUSTRISTNG agree well with observations at z=0.1, they are too shallow at z=1, predicting an evolution of =<0.5Gyr per decade in mass. Secondly, we consider the connections between green-valley, post-starburst, and quiescent galaxies, using our inferred SFH shapes and the distributions of galaxy physical properties on the UVJ diagram. The majority of our lowest-mass galaxies (M_*_~10^10.5^M_{sun}_) are consistent with formation in recent (z<2), intense starburst events, with time-scales of =<500Myr. A second class of objects experience extended star-formation epochs before rapidly quenching, passing through both green-valley and post-starburst phases. The most massive galaxies in our sample are extreme systems: already old by z=1, they formed at z~5 and quenched by z=3. However, we find evidence for their continued evolution through both AGN and rejuvenated star-formation activity.