Radar detection and communication are two of several sub-tasks essential for the operation of next-generation autonomous vehicles (AVs). The former is required for sensing and perception, more frequently so under various unfavorable environmental conditions such as heavy precipitation; the latter is needed to transmit time-critical data. Forthcoming proliferation of faster 5G networks utilizing mmWave is likely to lead to interference with automotive radar sensors, which has led to a body of research on the development of Joint Radar Communication (JRC) systems and solutions. This paper considers the problem of time-sharing for JRC, with the additional simultaneous objective of minimizing the average age of information (AoI) transmitted by a JRC-equipped AV. We formulate the problem as a Markov Decision Process (MDP) where the JRC agent determines in a real-time manner when radar detection is necessary, and how to manage a multiclass data queue where each class represents different urgency levels of data packets. Simulations are run with a range of environmental parameters to mimic variations in real-world operation. The results show that deep reinforcement learning allows the agent to obtain good results with minimal a priori knowledge about the environment.