The primary goal of this research is to obtain an approximate solution to the  Hirota-Satsuma coupled KdV equations with the lowest error rate, specifically when the time derivative is associated with a fraction and subject to specific initial conditions. In this work, two basic concepts are presented, the fractional Caputo derivative of order α>0 and the Sumudu Transform (ST), along with some inherent properties of this transformation. To address the problem at hand, a hybrid approach was adopted using numerical methods, combining the Sumudu Transform and the Adomian Decomposition Method (ADM). The Adomian Decomposition Method, as it is known, works to deal with complex nonlinear boundaries that are difficult to deal with directly. The article's methodology is carefully designed to prioritize clarity and simplicity and avoid unnecessary complexity. The results derived from this proposed method show a commendable level of accuracy and agree closely with exact solutions. The paper performs a comprehensive error analysis, ensuring the robustness and accuracy of the results.