Graphene-metal hybrid nanostructures have attracted considerable attention due to their potential applications in nanophotonics and optoelectronics. The output characteristics of devices based on such nanostructures largely depend on the properties of the metals. Here, we study the optical, electrical and structural properties of continuous thin gold and copper films grown by electron beam evaporation on monolayer graphene transferred onto silicon dioxide substrates. We find that the presence of graphene has a significant effect on optical losses and electrical resistance, both for thin gold and copper films. Furthermore, the growth kinetics of gold and copper films vary greatly; in particular, we found here a significant dependence of the properties of thin copper films on the deposition rate, unlike gold films. Our work provides new data on the optical properties of gold and copper, which should be considered in modeling and designing devices with graphene-metal nanolayers.