The atomic anharmonic vibration and the electron-phonon interaction are considered, and then a physical model about the metal-based epitaxial graphene is built. Variations of the electrical conductivity and the Fermi velocity with temperature for the metal-based epitaxial graphene are given based on the solid state physics theory or method. The alkali-metal epitaxial graphene is selected as the substrate, and then the influences of substrate material, electron-phonon interaction and the anharmonic vibration on the electrical conductivity and the Fermi velocity of epitaxial graphene are discussed. Some results are shown as follows. Firstly, at zero temperature, the electrical conductivity and the Fermi velocity of the alkali-metal-base epitaxial graphene increase with the number of the atoms in substrate material increasing. Secondly, the electrical conductivity of epitaxial graphene decreases with temperature rising. Furthermore, the variation rate also decreases with temperature rising. Generally, the electrical conductivity originates mainly from electrons and phones. The electronic contribution to the electrical conductivity varies with temperature slowly, but the phone contribution to electrical conductivity varies with temperature evidently. Therefore, the contribution of phonons to electrical conductivity is much larger than that of electrons. Furthermore, the contribution increases with the number of atoms in basal elements. The phonon contribution to conductivity decreases with temperature rising, but it is unrelated to the basal elements. Thirdly, the Fermi velocity of the epitaxial graphene increases with temperature slowly. The variation of the Fermi velocity with temperature decreases with the increase of interaction between the graphene and the basal atoms. However, it increases with the number of atoms of the basal materials. The anharmonic effect causes important influences on the electrical conductivity and the Fermi velocity. Under the harmonic approximation the velocity is constant. However, the conductance increases rapidly with temperature. With considering the atomic anharmonic terms, the Fermi velocity increases with temperature. The variation of the electrical conductivity with temperature increasing becomes slower. If the temperature is higher, the anharmonic effects become more evident.