We report on coarse-grained molecular dynamics simulations of the electroosmotic flow (EOF) of an electrolyte confined in a cylindrical, nanoscopic pore. We present results for the equilibrium distribution of fluid particles and ions in the electrolyte, and we show that our computational model reproduces the well-known characteristics of EOF in the steady-state regime, in particular the well-known pluglike character of this type of flow when the Debye length is small compared to the characteristic channel size. Upon adding a number of neutral, grafted polymer chains on the interior capillary surface, we find a significant reduction of the magnitude of the EOF. We characterize the polymer coatings and further show that the observed reduction in flow strength, as a function of polymer surface coverage, is in quantitative agreement with recent theoretical scaling predictions regarding the coupling of EOF and polymer coatings in small Debye length systems. As far as we know, our results constitute the first independent, quantitative verification of these predictions.