Metal-semiconductor interfaces play a crucial role not only for regulating the electronic conduction mechanism but also in determining new functionalities in nanosized devices. In this work, we reported the investigation of the junction properties of single ZnO nanowires (NWs) asymmetrically contacted by means of a Pt electrochemically inert and a Cu electrochemically active electrode. At low applied voltages, these devices operate as diodes where the conduction mechanism was found to be dominated by the Schottky barrier at the Cu/ZnO interface. Junction parameters such as the Schottky barrier height, the ideality factor and the series resistance have been analyzed according to the thermionic emission theory. Different methods for parameter retrieval from I-V-T measurements are discussed and compared. A potential fluctuation model is considered in order to account for barrier inhomogeneities, revealing the presence of two Gaussian distribution of barrier heights. On the other hand, new device features arise from electrochemical dissolution and migration of Cu ions along the NW when high electric fields are implied. These electrochemical processes are underlaying the resistive switching and memristive behavior observed in single ZnO NWs, as suggested also by direct observation of Cu nanoclusters along the nanostructures after the switching events.