Free-electron-like image potential states are observed in scanning tunneling spectroscopy on graphene quantum dots on Ir(111) acting as potential wells. The spectrum strongly depends on the size of the nanostructure as well as on the spatial position on top, indicating lateral confinement. Analysis of the substructure of the first state by the spatial mapping of the constant energy local density of states reveals characteristic patterns of confined states. The most pronounced state is not the ground state, but an excited state with a favorable combination of the local density of states and parallel momentum transfer in the tunneling process. Chemical gating tunes the confining potential by changing the local work function. Our experimental determination of this work function allows us to deduce the associated shift of the Dirac point.