In this work, we investigate and numerically approximate an initial-boundary value problem governed by Sobolev type differential equation exhibiting an initial layer. The problem is approached through a finite difference framework specifically designed to remain unaffected by the presence of a small perturbation parameter, ensuring robustness and stability. A novel hybrid numerical strategy is constructed by employing a non-polynomial trigonometric cubic B-spline (TCBS) collocation technique for the spatial discretization on a uniform partition, while time advancement is carried out through an implicit Euler procedure defined over a Shishkin mesh to effectively capture sharp solution gradients. The developed scheme guarantees parameter-uniform convergence, and its theoretical stability and error properties are rigorously analyzed. A set of numerical simulations is performed to validate the effectiveness of the method, demonstrating its high accuracy, consistency, and suitability for handling singularly perturbed pseudo-parabolic problems with initial layer.