We performed in situ magnetoresistance measurements in the ultrahigh-vacuum environment to reveal the quantum phase transitions of single unit-layer NbSe2 epitaxially grown on bilayer graphene. It was found that the superconductor-normal state transition caused by the surface-normal magnetic field was intermediated by a quasimetallic state. This behavior is consistent with the "Bose metal" picture where a finite dissipation is caused by the breaking of phase coherence due to strong gauge-field fluctuation. On one hand, around the mean-field critical temperature, the onset of transition from the normal state to the low-temperature states was governed by the amplitude-fluctuation effect, prominently under the magnetic field. We applied scaling theories to determine the phase boundaries. The result of scaling analyses revealed a complex but essential phase diagram of the single unit-layer NbSe2 as a two-dimensional superconductor.