A high-resolution X-ray spectroscopic observation was made of the RS CVn-type binary star HR 1099 using the Resolve instrument onboard XRISM for its calibration purposes. During the ∼400 ks telescope time covering 1.5 binary orbit, a flare lasting for ∼100 ks was observed with a released X-ray radiation energy of  ~10^34 erg, making it the first stellar flare ever observed with an X-ray microcalorimeter spectrometer. The flare peak count rate is 6.4 times higher than that in quiescence and is distinguished clearly in time thanks to the long telescope time. Many emission lines were detected in the 1.7–10 keV range both in the flare and quiescent phases. Using the high spectral resolution of Resolve in the Fe K band (6.5–7.0 keV), we resolved the inner-shell lines of Fe XIX– XXIV as well as the outer-shell lines of Fe XXV– XXVI. These lines have peaks in the contribution functions at different temperatures over a wide range, allowing us to construct the differential emission measure (DEM) distribution over the electron temperature of 1–10 keV (roughly 10–100 MK) based only on Fe lines, thus without an assumption of the elemental abundance. The DEM has a bimodal distribution, and only the hotter component increased during the flare. The elemental abundance was derived based on the DEM distribution thus constructed. A significant increase of Ca and Fe (elements of a low first ionization potential), but not of Si, S, and Ar (those with a high potential), was observed during the flare. This is a behavior seen in some giant solar flares and the present result is a clear example in stellar flares.