In this paper, the scheduling of downlink file transmission in one cell with the assistance of cache nodes with finite cache space is studied. Specifically, requesting users arrive randomly and the base station (BS) reactively multicasts files to the requesting users and selected cache nodes. The latter can offload the traffic in their coverage areas from the BS. We consider the joint optimization of the abovementioned file placement and delivery within a finite lifetime subject to the cache space constraint. Within the lifetime, the allocation of multicast power and symbol number for each file transmission at the BS is formulated as a dynamic programming problem with a random stage number. Note that there are no existing solutions to this problem. We develop an asymptotically optimal solution framework by transforming the original problem to an equivalent finite-horizon Markov decision process (MDP) with a fixed stage number. A novel approximation approach is then proposed to address the curse of dimensionality, where the analytical expressions of approximate value functions are provided. We also derive analytical bounds on the exact value function and approximation error. The approximate value functions depend on some system statistics, e.g., requesting users' distribution. One reinforcement learning algorithm is proposed for the scenario where these statistics are unknown.