This article presents a robust optimal consensus control strategy for switched nonlinear multi-agent systems (MASs). The robust control problem with unmatched uncertainties is first reformulated as an optimal control framework through the introduction of augmented control inputs and a tailored cost function. To construct a distributed integral sliding-mode control (ISMC), a novel measurement error term is designed to enforce finite-time convergence of the consensus error to the origin. An event-based mechanism is incorporated to minimize controller execution, thereby optimizing computational and communication resource utilization. A concurrent learning approach is proposed to update neural network (NN) weights, where the value function is approximated by a critic NN. This method circumvents the constraints of initial admissible control and persistent excitation (PE) conditions inherent in traditional adaptive dynamic programming (ADP) designs. Theoretical analysis demonstrates that the developed event-triggered ADP controller guarantees system robustness for nonlinear MASs and ensures the uniform ultimate boundedness (UUB) of critic NN weight estimation errors. Finally, simulation results are presented to validate the effectiveness of the proposed control methodology.