We explore the variation in stellar population ages for Coma cluster galaxies as a function of projected cluster-centric distance, using a sample of 362 red-sequence galaxies with high signal-to-noise ratio spectroscopy. The sample spans a wide range in luminosity (0.02–4 L*) and extends from the cluster core to near the virial radius. We find a clear distinction in the observed trends of the giant and dwarf galaxies. The ages of red-sequence giants are primarily determined by galaxy mass, whether parametrized by velocity dispersion, luminosity, stellar mass or dynamical mass, with only weak modulation by environment, in the sense that galaxies at larger cluster-centric distance are slightly younger. For red-sequence dwarfs (with mass ≲ 1010 M⊙), the roles of mass and environment as predictors of age are reversed: there is little dependence on mass, but strong trends with projected cluster-centric radius are observed. The average age of dwarfs at the 2.5 Mpc limit of our sample is approximately half that of dwarfs near the cluster centre. The gradient in dwarf galaxy ages is a global cluster-centric trend, and is not driven by the ongoing merger of the NGC 4839 group to the south west of Coma. We interpret these results using environmental histories extracted from the Millennium Simulation for members of massive clusters. Hierarchical cluster assembly naturally leads to trends in the accretion times of galaxies as a function of projected cluster-centric radius. On average, simulated galaxies now located in cluster cores joined haloes above any given mass threshold earlier than those now in the outskirts of clusters. We test environmental quenching models, in which star formation is halted in galaxies when they enter haloes of a given mass, or when they become satellites. Our models broadly reproduce the gradients observed in Coma, but for dwarf galaxies the efficiency of environmental quenching must be very high to match the strong trends observed.