We present the first high spectral resolution X-ray observation of the giant elliptical galaxy NGC 4636, obtained with the Reflection Grating Spectrometer on-board the XMM-Newton Observatory. The resulting spectrum contains a wealth of emission lines from various charge states of oxygen, neon, magnesium, and iron. Examination of the cross-dispersion profiles of several of these lines provides clear, unambiguous evidence of resonance scattering by the highest oscillator strength lines, as well as a weak temperature gradient in the inner regions of the interstellar medium. We invoke a sophisticated new Monte Carlo technique which allows us to properly account for these effects in performing quantitative fits to the spectrum. Our spectral fits are not subject to many of the systematics that have plagued earlier investigations. The derived metal abundances are higher than have been inferred from prior, lower spectral resolution observations of this source (Awaki et al. 1994), but are still incompatible with conventional chemical enrichment models of elliptical galaxies. In addition, our data are incompatible with standard cooling flow models for this system-our derived upper limit to the mass deposition rate is below the predicted value by a factor of 3-5.