In a Fog Radio Access Network (F-RAN) architecture, edge nodes (ENs), such as base stations, are equipped with limited-capacity caches, as well as with fronthaul links that can support given transmission rates from a cloud processor. Existing information-theoretic analyses of content delivery in F-RANs have focused on offline caching with separate content placement and delivery phases. In contrast, this work considers an online caching set-up, in which the set of popular files is time-varying and both cache replenishment and content delivery can take place in each time slot. The analysis is centered on the characterization of the long-term Normalized Delivery Time (NDT), which captures the temporal dependence of the coding latencies accrued across multiple time slots in the high signal-to-noise ratio regime. Online caching and delivery schemes based on reactive and proactive caching are investigated, and their performance is compared to optimal offline caching schemes both analytically and via numerical results.