This paper investigates an air-ground integrated multi-access edge computing system, which is deployed by an infrastructure provider (InP). Under a business agreement with the InP, a third-party service provider provides computing services to the subscribed mobile users (MUs). MUs compete for the shared spectrum and computing resources over time to achieve their distinctive goals. From the perspective of an MU, we deliberately define the age of update to capture the staleness of information from refreshing computation outcomes. Given the system dynamics, we model the interactions among MUs as a stochastic game. In the Nash equilibrium without cooperation, each MU behaves in accordance with the local system states and conjectures. We can hence transform the stochastic game into a single-agent Markov decision process. As another major contribution, we develop an online deep reinforcement learning (RL) scheme that adopts two separate double deep Q-networks to approximate the Q-factor and the post-decision Q-factor, respectively. The deep RL scheme allows each MU to optimize the behaviours with unknown dynamic statistics. Numerical experiments show that our proposed scheme outperforms the baselines in terms of the average utility under various system conditions.