Interaction of sodium with graphene (Gr) on Ir(111) was studied with the aim to resolve the issue of Na adsorption/intercalation kinetics. The system Na/Gr/Ir(111) was studied by means of angle-resolved photoemission spectroscopy, low-energy electron diffraction, and ab initio density functional theory (DFT) calculation. It has been found that at room temperature (RT) and low concentrations Na is dominantly adsorbed on graphene. At higher concentrations, an intercalation process sets in so that it is possible to observe the coexistence of these two states. Eventually, all Na atoms are found in the intercalated state as determined by exposure to oxygen. While adsorption of Na on graphene already intercalated by Na [Na/Gr/Na/Ir(111) system] at RT was not possible, we could observe Li adsorption through the increase of Dirac point binding energy. Li coadsorption strongly affects the binding energy of the iridium surface state as well. This finding was supported by DFT calculations of adsorption energy of Na and Li on bare and fully Na intercalated graphene.