We consider a dynamic content caching problem wherein the contents get updated at a central server, and local copies of a subset of contents are cached at a local cache associated with a Base station (BS). When a content request arrives, based on whether the content is in the local cache, the BS can decide whether to fetch the content from the central server or serve the cached version from the local cache. Fetching a content incurs a fixed fetching cost, and serving the cached version incurs an ageing cost proportional to the age-of-version (AoV) of the content. The BS has only partial information regarding AoVs of the contents. We formulate an optimal content fetching and caching problem to minimize the average cost subject to cache capacity constraints. The problem suffers from the curse of dimensionality and is provably hard to solve. We formulate this problem as a continuous time restless multi-armed bandit process (RMAB), where a single content problem of the corresponding RMAB is a partially observable Markov decision process. We prove the indexability of the single content problem, and provide a Whittle index based solution to this problem. Finally, we compare the performance with recent work and show that our proposed policy is optimal via simulations. We demonstrate that in Fig 1, how the policy performs better than the existing works and also performs close to the optimal. We further introduce “wait” option for the BS. In particular, the BS either can serve the local version if available or can fetch a fresh version or can wait for additional requests before fetching and serving a fresh version. In addition to fetching and ageing costs, waiting costs can incur for each waiting requests. We focus on the optimal actions subject to the cache capacity constraint at each decision epoch, aiming at minimizing the long term average cost. Allowing “wait” option adds the queue lengths, to the system’s state. It also substantially changes the state dynamics, the analysis and the caching policy. We again propose Whittle index based policy as a heuristic to this problem. We show that having a “wait” option improves the average cost. We demonstrate this in In Fig 2.