An apparatus was developed to obtain, for the first time, 2D maps of the surface resistance of the inner surface of an operating superconducting radio-frequency niobium cavity by a low-temperature laser scanning microscopy technique. This allows identifying non-uniformities of the surface resistance with a spatial resolution of about 2.4 mm and surface resistance resolution of ~1 μΩ at 3.3 GHz. A signal-to-noise ratio of about 10 dB was obtained with 240 mW laser power and 1 Hz modulation frequency. The various components of the apparatus, the experimental procedure and results are discussed in detail in this contribution.