We demonstrated high-sensitivity optical spectral imaging of a single layer of a quantum gas of ytterbium atoms in a two-dimensional optical lattice using the ultranarrow S-1(0)-P-3(2) transition. We successfully obtained a set of excitation-frequency-dependent fluorescence images with an excitation laser of the linewidth of 1 kHz (FWHM), and the overall features were well explained by considering the inhomogeneous light shift originating from the Gaussian beam shape of the optical lattice potential which provided the steepest potential gradient of 3.6 kHz/mu m. This result is also the successful demonstration of the tunable local atom addressing along the equipotential contour depending on the excitation laser frequency with the frequency resolution of 8 kHz and the spatial resolution of approximately 2 mu m. The demonstrated technique will be useful for many purposes including the measurement of interaction shift in the study of a quantum gas and quantum information processing.