$^{13}C$ NMR (nuclear magnetic resonance) measurements have been performed to investigate the local electronic structure of a superconducting graphite intercalation compound $CaC_6$ ($T_c$ = 11.4 K). A large number of single crystals were stacked and sealed in a quartz tube for naturally abundant $^{13}C$ NMR. The spectrum, Knight shift, linewidth, and spin-lattice relaxation time $T_1$ were measured in the normal state as a function of temperature down to 80 K at 8.0 T perpendicular to the c-axis. The $^{13}C$ NMR spectrum shows a single narrow peak with a very small Knight shift. The Knight shift and the linewidth of the $^{13}C$ NMR are temperature-independent around, respectively, +0.012% and 1.2 kHz. The spin-lattice relaxation rate, $1/T_1$, is proportional to temperature confirming a Korringa behavior as for non-magnetic metals. The Korringa product is measured to be $T_1T;=;210;s{cdot}K$. From this value, the Korringa ratio is deduced to be $xi$ = 0.73, close to unity, which suggests that the independent-electron description works well for $CaC_6$, without complications arising from correlation and many-body effects.