Understanding shear flow behavior in compressible, viscous, micropolar real gases is essential for both theoretical advancements and practical engineering applications. This study develops a comprehensive model that integrates micropolar fluid theory with compressible flow dynamics to accurately describe the behavior of real gases under shear stress. We formulate the governing equations by incorporating viscosity and micropolar effects and transform the obtained system into the mass Lagrangian coordinates. Two numerical methods, Faedo–Galerkin approximation and finite-difference methods, are used to solve it. These methods are compared using several benchmark examples to assess their accuracy and computational efficiency. Both methods demonstrate good performance, achieving equally precise results in capturing essential flow characteristics. However, the finite-difference method offers advantages in speed, stability, and lower computational demands. This research bridges gaps in existing models and establishes a foundation for further investigations into complex flow phenomena in micropolar real gases.