The reaction of hydrogen with oxygen adsorbed on an Ru(1010) surface has been studied by density-functional calculations and kinetic Monte Carlo simulations. In agreement with experiment, it has been found that molecular hydrogen does not react with adsorbed O. In contrast, the hydrogenation of oxygen by an atomic H beam occurs spontaneously and results in the formation of adsorbed OH molecules. Subsequent impinging H-atoms can either initiate the formation of water, which readily desorbs at room temperature thus removing the O from the surface, or lead to formation and desorption of H2. It is the latter channel that hinders, at 300 K, a complete removal of O from Ru capping layers on Si/Mo mirrors for extreme ultraviolet radiation. The estimated height of the barrier for the Langmuir-Hinshelwood reaction between adsorbed H and OH, 0.92 eV, and related position of the H2O peak in model desorption spectra (approximately 320 K) are consistent with recent experiments. The H2 desorption peak appears at higher temperature, approximately 350 K, so that in the range from 320 to 330 K adsorbed hydrogen atoms will react predominantly with OH. Hence, the present simulations predict that an efficient removal of the chemisorbed O from Ru capping layers can be achieved by heating the surface to 320-330 K in a molecular hydrogen atmosphere.