Ultrafast pulses, particularly those with durations under 100 fs, are crucial in achieving unprecedented precision and control in light-matter interactions. However, conventional on-chip photonic platforms are not inherently designed for ultrafast time-domain operations, posing a significant challenge in achieving essential parameters such as high peak power and high temporal resolution. This challenge is particularly pronounced when propagating through integrated waveguides with nonlinear and high-dispersion profiles. In addressing this challenge, we present a design methodology for ultrafast pulse propagation in dispersive integrated waveguides, specifically focused on enhancing the time-domain characteristics of one-dimensional grating waveguides (1DGWs). The proposed methodology aims to determine the optimal structural parameters for achieving maximum peak power, enhanced temporal resolution, and extended pulse storage duration during ultrafast pulse propagation. To validate this approach, we design and fabricate two specialized 1DGWs on a silicon-on-insulator (SOI) platform. A digital finite impulse response (FIR) model, trained with both transmission and phase measurement data, is employed to obtain ultrafast time-domain characteristics, enabling easy extraction of these results. Our approach achieves a 2.8-fold increase in peak power and reduces pulse broadening by 24 %, resulting in a smaller sacrifice in temporal resolution. These results can possibly pave the way for advanced light-matter interactions within dispersive integrated waveguides.