The concept of age of information (AoI) has been proposed to quantify information freshness, which is crucial for time-sensitive applications. However, in millimeter wave (mmWave) communication systems, the link blockage caused by obstacles and the severe path loss greatly impair the freshness of information received by the user equipments (UEs). In this paper, we focus on reconfigurable intelligent surface (RIS)-assisted mmWave communications, where beamforming is performed at transceivers to provide directional beam gain and a RIS is deployed to combat link blockage. We aim to maximize the system sum rate while satisfying the information freshness requirements of UEs by jointly optimizing the beamforming at transceivers, the discrete RIS reflection coefficients, and the UE scheduling strategy. To facilitate a practical solution, we decompose the problem into two subproblems. For the first per-UE data rate maximization problem, we further decompose it into a beamforming optimization subproblem and a RIS reflection coefficient optimization subproblem. Considering the difficulty of channel estimation, we utilize the hierarchical search method for the former and the local search method for the latter, and then adopt the block coordinate descent (BCD) method to alternately solve them. For the second scheduling strategy design problem, a low-complexity heuristic scheduling algorithm is designed. Simulation results show that the proposed algorithm can effectively improve the system sum rate while satisfying the information freshness requirements of all UEs.