We experimentally investigate a weakly disordered one-dimensional Anderson-localizing lasing system in the neighborhood of the critical disorder. This critical disorder δcr separates two regimes, in one of which the transport parameters show a wavelength dependence, but not in the other. We first theoretically study the temporal characteristics of lasing pulses, and extract the lifetimes of the lasing modes. The distribution of the lifetimes shows a bimodal distribution under the δcr, which condenses into a monomodal distribution beyond the δcr. Next, we implement an experimental scheme using a linear array of microresonators with gain, which sustains Anderson localization. Emission pulses were analyzed on a streak camera to build lifetime distributions in the experimental samples, which reproduced the theoretical predictions to a very good degree. These observations provide a direct demonstration of the effect of critical disorder on Anderson-localizing lasing systems.