This research confronts the intricate challenges posed by nonlinear dynamics in spacecraft optimal control through the application of data-driven methodologies to approximate the Koopman operator. The Koopman operator, characterized by infinite dimensions, facilitates the linear advancement of nonlinear state dynamics in a lifted space. However, developing an appropriate Koopman embedding function remains a challenging endeavor. To address the intricacies associated with the design of a suitable Koopman embedding function, we employ a Deep Neural Network (DNN) learning paradigm for the adaptive acquisition of the function basis and propagation matrices. Subsequently, the Model Predictive Control (MPC) is deployed in the linear embedding space, effectively leveraging the advantages offered by the Koopman operator. Simulation results substantiate the efficacy of the proposed strategy, affirming its capability in effectively identifying the intricate nonlinear dynamics within the spacecraft attitude control system.