A prototype for a highly sensitive, automated measurement system for the microwave surface resistance of high-$T_c$ superconductor films was set up, and tested by measuring the microwave surface resistances of high-$T_c$ $YBa_2Cu_3O_{7-delta}$ (YBCO) films at the frequency of about 19.6 GHz and the temperature of 30 K $sim$ 90 K. An open-ended $TE_{011}$ mode sapphire-loaded cylindrical cavity resonator was used as the measurement probe, where YBCO films were used as the endplates of the cylindrical cavity. The characteristics of the measurement system include functions to display the unloaded Q and the resonant frequency of the $TE_{011}$ mode resonator as well as the microwave surface resistance of the YBCO films, all simultaneously as a function of temperature. Applicability of the measurement system for investigating the homogeneity in the microwave properties of large high-$T_c$ superconductor films is discussed.