In the mechanism of two-dimensional (2D) superconductivity in doped graphene, it was predicted that the presence of dopant bands plays a key role. However, it has been challenging to fabricate an ordered phase of alkali metals on graphene owing to its van-der-Waals nature. We systematically study the phase formation of Rb on graphene epitaxially grown on H-passivated SiC(0001). We found a range of control parameters that stabilize the well-ordered 2 x 2 phase (RbC8), as confirmed by in-situ low-energy electron diffraction. Angle-resolved photoemission (ARPES) spectra taken from the 2 x 2 phase show a folded band of graphene and a parabolic band of Rb. In the vicinity of the Fermi level, the self-energy extracted from ARPES data reveals a clear signature of electronic coupling to the in-plane and out-of-plane phonon modes of C and Rb atoms. The electronic coupling of graphene to the in-plane vibration mode of Rb atoms is identified as a key factor for the enhancement of electron-phonon coupling.