Herein, the quantum transport in a graphene-based normal layer/insulator/p-wave superconductor tunnel junctions is studied by an extended Blonder-Tinkham-Klapwijk theory. It is shown that at the insulating region width being finite, the conductance and shot noise spectra exhibit oscillations as a function of applied bias voltage, which is a striking novel feature and absent in thin-barrier limit. Particularly, the oscillatory behaviors between the px-, py-, and px+ipy-wave pairing symmetries are significantly different, stemming from the different types of zero-energy bound states. These features could not only be experimentally used to identify the p-wave superconductivity of graphene but also pave the way to the future preparation of graphene p-wave superconducting devices.