The exchange bias (EB) is a unidirectional anisotropy in ferro-/antiferromagnetic (FM/AFM) systems which primarily leads to a horizontal shift of the magnetic hysteresis loop after field-cooling through the Néel temperature of the AFM. After being discovered in Co/CoO core-shell nanoparticles, it is nowadays investigated in diverse material systems. Co/CoO thin film systems are among those most widely studied. Here, we report measurements at different temperatures, below and above the Néel temperature of CoO, for different sample orientations with respect to the external magnetic field. The samples were produced by a molecular beam epitaxy growth of 8 nm Co on CoO(100) substrates after the substrates had been irradiated by heavy ions (xenon or uranium) to induce defects in the AFM. While the EB was relatively small, measurements of the bulk magnetization at low temperatures revealed unusually shaped hysteresis loops. Measurements of the magneto-optic Kerr effect, on the other hand, in which laser light interacts only with the surface magnetic layer, showed simple, nearly rectangular hysteresis loops. The transverse magnetization component was examined by changing the laser’s linear polarization direction. In this way, differences between magnetization reversal processes in xenon and uranium-irradiated samples could be shown. This study underlines the possibility to optically detect the magnetization at a sample surface, as opposed to other techniques which measure the bulk magnetization, and thus the advantage of complementary information on surface and bulk magnetization from optical and non-optical measurement methods.