In this work, we investigate the local effects of a single short-range impurity on the electron system of a short ballistic graphene Josephson Junction. Within the Dirac-Bogoliubov-De Gennes approach, we systematically analyze the local density states, whose subgap energy dependence enables us to distinguish between elastic and inelastic scattering processes and identify the magnetic nature of the impurity. Furthermore, we observe that the spatial dependence of the local density of states is a sensitive probe of the microscopic processes resulting in subgap impurity-induced bound states. The Fourier analysis evidences the wavevectors related to the momenta of the high transmissive channels in ballistic graphene.