This paper studies a finite-horizon Markov decision problem with information-theoretic constraints, where the goal is to minimize directed information from the controlled source process to the control process, subject to stage-wise cost constraints, aiming for an optimal control policy. We propose a new way of approximating a solution for this problem, which is known to be formulated as an unconstrained MDP with a continuous information-state using Q-factors. To avoid the computational complexity of discretizing the continuous information-state space, we propose a truncated rollout-based backward-forward approximate dynamic programming (ADP) framework. Our approach consists of two phases: an offline base policy approximation over a shorter time horizon, followed by an online rollout lookahead minimization, both supported by provable convergence guarantees. We supplement our theoretical results with a numerical example where we demonstrate the cost improvement of the rollout method compared to a previously proposed policy approximation method, and the computational complexity observed in executing the offline and online phases for the two methods.