Recently studied hyperbolic materials host unique phonon-polariton (PP) modes. The ultrashort wavelengths of these modes, as well as their low damping, hold promise for extreme subdiffraction nanophotonics schemes. Polar hyperbolic materials such as hexagonal boron nitride can be used to realize long-range coupling between PP modes and extraneous charge degrees of freedom. The latter, in turn, can be used to control and probe PP modes. Here we analyze coupling between PP modes and plasmons in an adjacent graphene sheet, which opens the door to accessing PP modes by angle-resolved photoemission spectroscopy (ARPES). A rich structure in the graphene ARPES spectrum due to PP modes is predicted, providing a new probe of PP modes and their coupling to graphene plasmons.