In integrated sensing and communication (ISAC) systems, the timeliness performance is pivotal to support high-precision sensing and reliable high-rate data communication, particularly for real-time applications. However, there are two critical challenges in the research on the timeliness of ISAC systems: joint characterization of sensing-communication time-liness tradeoffs and unified resource allocation optimization. To address these challenges, a timeliness-centric analysis and optimization framework is proposed for ISAC systems. First, the age of information (AoI) of sensing tasks and the delay of communication tasks are analyzed to evaluate the timeliness performance of ISAC systems. Then, the closed-form expressions of the outage probabilities for sensing and communication tasks are derived, quantifying their reliability under dynamic channel conditions. Next, a joint sensing-and-communication time-allocation optimization problem is established to minimize the weighted sum of sensing AoI and communication delay. To tackle the non-linear integer programming problem, a dynamic programming (DP) algorithm is designed, which can be verified to converge to an optimal solution with polynomial computational complexity. Simulation results are presented to validate the accuracy of the analytical results, and demonstrate the optimality and performance gains of our proposed DP algorithm.