Piecewise constant priors are routinely used in the Bayesian Cox proportional hazards model for survival analysis. Despite its popularity, large sample properties of this Bayesian method are not yet well understood. This work provides a unified theory for posterior distributions in this setting, not requiring the priors to be conjugate. We first derive contraction rate results for wide classes of histogram priors on the unknown hazard function and prove asymptotic normality of linear functionals of the posterior hazard in the form of Bernstein--von Mises theorems. Second, using recently developed multiscale techniques, we derive functional limiting results for the cumulative hazard and survival function. Frequentist coverage properties of Bayesian credible sets are investigated: we prove that certain easily computable credible bands for the survival function are optimal frequentist confidence bands. We conduct simulation studies that confirm these predictions, with an excellent behavior particularly in finite samples, showing that even simplest possible Bayesian credible bands for the survival function can outperform state-of-the-art frequentist bands in terms of coverage.