The challenge of modeling low-speed rarefied gas flow in the transition regime is well known. In this paper, we propose a numerical solution procedure for the regularized 13 moment equations within a finite-volume framework. The stress and heat flux equations arising in the method of moments are transformed into the governing equations for the stress and heat flux deviators based on their first-order approximation. To model confined flows, a complete set of wall boundary conditions for the 13 moment equations are derived based on the Maxwell wall-boundary model. This has been achieved by expanding the molecular distribution function to fourth-order accuracy in Hermite polynomials. Empirical correction factors are introduced into the boundary conditions and calibrated against direct simulation Monte Carlo data. The numerical predictions obtained from the regularized 13 moment equations and the Navier-Stokes-Fourier equations are compared with data generated using the direct simulation Monte Carlo method for planar Couette flow. For a range of wall velocities and Knudsen numbers (0.012-1.0), the results indicate that the regularized 13 moment equations are in good qualitative agreement with the direct simulation Monte Carlo data. The results also highlight limitations that are caused by the use of a first-order expansion of the third moment.