To achieve reliable and seamless content delivery services in connected vehicles, it is imperative to gather and employ road environment information through interactions with the surrounding infrastructure. Furthermore, caching the collected information enables rapid utilization within connected vehicles. For this objective, we propose a scenario that leverages infrastructure components such as road side units (RSUs) and macro base stations (MBSs) to cache road environment information. In this context, the age-of-information (AoI) concept serves as a critical factor for ensuring data freshness in ultra-fast time-varying road environments. Adhering to the AoI concept, it is crucial to maintain pertinent content in RSUs, update it prior to expiration, and distribute it to the targeted user vehicles (UVs). To address this goal, we introduce a two-stage algorithm composed of AoI-aware content caching and delay-aware content delivery, which are ultimately re-formulated using the Markov decision process (MDP) and Lyapunov optimization to ensure optimal decision-making in discrete time units. This strategy aims to enhance content delivery performance by minimizing AoI values in MBSs, RSUs, and UVs. Finally, data-intensive performance evaluation verifies that our proposed algorithm outperforms the other comparison algorithms.