Digital receivers based on 1-bit quantization and temporal oversampling w. r. t. the transmit signal bandwidth are a promising solution for the design of energy-efficient communications systems in the millimeter-wave (mmWave) and sub-terahertz bands. However, off-the-shelf algorithms for channel estimation cannot be applied as 1-bit quantization is a highly non-linear operation. Phase noise (PN) in particular has a deteriorating effect on the communication performance at these high frequencies and, therefore, needs to be tracked and compensated at the receiver. In this context, we derive an analytical solution for a close approximation of the Bayesian Cramér-Rao bound for PN estimation in systems employing 1-bit quantization, which provides insights into the impact of various design parameters on the achievable estimation performance. Furthermore, we use the bound to benchmark the performance of two existing PN estimators, showing that one of these estimators performs close to the optimum.