Contacting interfaces with physical isolation and weak interactions usually act as barriers for electrical conduction. The electrical contact conductance across interfaces has long been correlated with the true contact area or the "contact quantity". Much of the physical understanding of the interfacial electrical contact quality was primarily based on Landauer's theory or Richardson formulation. However, a quantitative model directly connecting contact conductance to interfacial atomistic structures still remains absent. Here, we measure the atomic-scale local electrical contact conductance instead of local electronic surface states in graphene/Ru(0001) superstructure, via atomically resolved conductive atomic force microscopy. By defining the "quality" of individual atom-atom contact as the carrier tunneling probability along the interatomic electron transport pathways, we establish a relationship between the atomic-scale contact quality and local interfacial atomistic structure. This real-space model unravels the atomic-level spatial modulation of contact conductance, and the twist angle-dependent interlayer conductance between misoriented graphene layers.