With the proliferation of latency-critical applications, fog-radio network (FRAN) has been envisioned as a paradigm shift enabling distributed deployment of cloud-clone facilities at the network edge. In this paper, we consider proactive caching for a one-user one-access point (AP) fog computing system over a finite time horizon, in which consecutive tasks of the same type of application are temporarily correlated. Under the assumption of predicable length of the task-input bits, we formulate a long-term weighted-sum energy minimization problem with three-slot correlation to jointly optimize computation offloading policies and caching decisions subject to stringent per-slot deadline constraints. The formulated problem is hard to solve due to the mixed-integer non-convexity. To tackle this challenge, first, we assume that task-related information are perfectly known {\em a priori}, and provide offline solution leveraging the technique of semi-definite relaxation (SDR), thereby serving as theoretical upper bound. Next, based on the offline solution, we propose a sliding-window based online algorithm under arbitrarily distributed prediction error. Finally, the advantage of computation caching as well the proposed algorithm is verified by numerical examples by comparison with several benchmarks.