We performed coordinated observations of AR 12205, which produced a C-class flare on 2014 November 11, with the Interface Region Imaging Spectrograph (IRIS) and the Domeless Solar Telescope (DST) at Hida Observatory. Using spectral data in the Si IV 1403 AA, C II 1335 AA, and Mg II h and k lines from IRIS and the Ca II K, Ca II 8542 AA, and H$alpha$ lines from DST, we investigated a moving flare kernel during the flare. In the Mg II h line, the leading edge of the flare kernel showed the intensity enhancement in the blue wing, and the smaller intensity of the blue-side peak (h2v) than that of the red-side one (h2r). The blueshift lasted for 9-48 s with a typical speed of 10.1 $pm$ 2.6 km s$^{-1}$ and it was followed by the high intensity and the large redshift with a speed of up to 51 km s$^{-1}$ detected in the Mg II h line. The large redshift was a common property for all six lines but the blueshift prior to it was found only in the Mg II lines. A cloud modeling of the Mg II h line suggests that the blue wing enhancement with such peak difference can be caused by a chromospheric-temperature (cool) upflow. We discuss a scenario in which an upflow of cool plasma is lifted up by expanding hot plasma owing to the deep penetration of non-thermal electrons into the chromosphere. Furthermore, we found that the blueshift persisted without any subsequent redshift in the leading edge of the flare kernel during its decaying phase. The cause of such long-lasting blueshift is also discussed.