This article focuses on multi-agent distributed optimization problems with a common decision variable, a global linear equality constraint, and local set constraints over directed interconnection topologies. We propose a novel ADMM based distributed algorithm to solve the above problem. During every iteration of the algorithm, each agent solves a local convex optimization problem and utilizes a finite-time ``approximate'' consensus protocol to update its local estimate of the optimal solution. The proposed algorithm is the first ADMM based algorithm with convergence guarantees to solve distributed multi-agent optimization problems where the interconnection topology is directed. We establish two strong explicit convergence rate estimates for the proposed algorithm to the optimal solution under two different sets of assumptions on the problem data. Further, we evaluate our proposed algorithm by solving two non-linear and non-differentiable constrained distributed optimization problems over directed graphs. Additionally, we provide a numerical comparison of the proposed algorithm with other state-of-the-art algorithms to show its efficacy over the existing methods in the literature.