This letter describes a robust distributed inverse optimal control framework for a multi-agent discrete-time nonlinear system, where the dynamics of each agent is directly affected by terms that depend on the state and input of the neighborhood agents and other disturbance signals. The individual local cost is formulated and a control solution for each agent is derived considering an inverse optimal control approach. To address the interaction between the agents, a coordination method based on a non-cooperative game is proposed. Using Lyapunov and Input-to-State Stability (ISS) arguments, we derive conditions under which the proposed game converges to a fixed point and the overall multi-agent system is ISS with respect to the disturbance signals. Simulation results for a coupled pendula system are presented.