A high-order hybrid numerical framework is developed by coupling a three-stage exponential time integrator with a Runge–Kutta scheme for the efficient solution of partial differential equations involving first-order time derivatives. The proposed scheme attains third-order temporal accuracy and is rigorously validated through stability and convergence analyses for both scalar and coupled systems. Its effectiveness is demonstrated by simulating unsteady Eyring-Prandtl non-Newtonian nanofluid flow over a Riga plate with coupled heat and mass transfer under electromagnetic actuation. The physical model accounts for Brownian motion and thermophoresis, and the nanofluid considered is a Prandtl-type non-Newtonian base fluid containing suspended nanoparticles, with heat and mass transport governed by coupled momentum, energy, and concentration equations. Numerical simulations are performed over practically relevant parameter ranges, with the Reynolds number fixed at $Re=5$ and the Prandtl number set to $Pr=3$ to represent moderate inertial and thermal diffusion effects typical of nanofluid transport systems. To enhance computational efficiency, an artificial neural network (ANN)-based surrogate model is developed to predict the skin friction coefficient and local Sherwood number as functions of Reynolds number, Prandtl number, Schmidt number, Brownian motion, and thermophoresis parameters. The training dataset is generated entirely from high-fidelity numerical simulations produced by the proposed hybrid scheme. The data are systematically partitioned into 70% for training, 15% for validation, and 15% for testing, ensuring reliable generalization. Regression analysis yields a near-unity correlation coefficient ( $R\approx 0.99$), while error histograms exhibit tightly clustered residuals around zero, confirming high predictive accuracy. Furthermore, a benchmark convergence study using Stokes’ first problem demonstrates that the proposed scheme consistently achieves lower global error norms than the classical Runge–Kutta method for identical spatial and temporal resolutions. Overall, this study introduces a novel computational intelligence framework that integrates high-order numerical solvers with machine learning, offering a robust and time-efficient tool for advanced modeling and real-time prediction of non-Newtonian nanofluid transport phenomena under electromagnetic flow control.