In this paper, by incorporating the network-based event-triggered formulation, the robust adaptive critic control design for a class of nonlinear continuous-time systems is investigated to fulfill disturbance rejection. First, the designed problem with output information is formulated as a two-player zero-sum differential game and the adaptive critic mechanism is employed toward the event-based minimax optimization involving a suitable triggering condition. Then, the event-based optimal control law and the time-based worst-case disturbance law are learned by training the critic neural network. Besides, the closed-loop system is constructed with stability proof of the critic error dynamics and the sampled-data plant. The theoretical analysis has demonstrated that the infamous Zeno behavior of the proposed event-based adaptive critic design has been avoided. Finally, the developed method is applied toward the robot arm plant, as a mechanical component of the complex robot system, so as to substantiate the performance of disturbance rejection.