The combined influence of linear damping and noise on a dynamical finite-time-singularity model is considered for a single degree of freedom. The noise resolves the finite-time-singularity and replaces it by a first-passagetime distribution with a peak at the singularity and a long time tail. The damping introduces a characteristic cross-over time. In the early time regime the first-passage-time distribution shows a power law behavior with scaling exponent depending on the ratio of the non linear coupling strength to the noise strength. In the late time regime the damping prevails. The study might be of relevance in the context of hydrodynamics on a nanometer scale, in material physics, and in biophysics.