Magnetically actuated fish-like robots offer promising solutions for underwater exploration due to their miniaturization and agility; however, precise control remains a significant challenge because of nonlinear fluid dynamics, flexible fin hysteresis, and the variable-duration control steps inherent to the actuation mechanism. This paper proposes a comprehensive data-driven control framework to address these complexities without relying on analytical modeling. Our methodology comprises three core components: 1) developing a forward dynamics model (FDM) using a neural network trained on real-world experimental data to capture state transitions under varying time steps; 2) integrating this FDM into a gradient-based model predictive control (G-MPC) architecture to optimize control inputs for path following; and 3) applying imitation learning to approximate the G-MPC policy, thereby reducing the computational cost for real-time implementation. We validate the approach through simulations utilizing the identified dynamics model. The results demonstrate that the G-MPC framework achieves accurate path convergence with minimal root mean square error (RMSE), and the imitation learning controller (ILC) effectively replicates this performance. This study highlights the potential of data-driven control strategies for the precise navigation of miniature, fish-like soft robots.