We have developed a system combining a back-illuminated Complementary-Metal-Oxide-Semiconductor (CMOS) imaging sensor and Xilinx Zynq System-on-Chip (SoC) device for a soft X-ray (0.5-10 keV) imaging spectroscopy observation of the Sun to investigate the dynamics of the solar corona. Because typical timescales of energy release phenomena in the corona span a few minutes at most, we aim to obtain the corresponding energy spectra and derive the physical parameters, i.e., temperature and emission measure, every few tens of seconds or less for future solar X-ray observations. An X-ray photon-counting technique, with a frame rate of a few hundred frames per second or more, can achieve such results. We used the Zynq SoC device to achieve the requirements. Zynq contains an ARM processor core, which is also known as the Processing System (PS) part, and a Programmable Logic (PL) part in a single chip. We use the PL and PS to control the sensor and seamless recording of data to a storage system, respectively. We aim to use the system for the third flight of the Focusing Optics Solar X-ray Imager (FOXSI-3) sounding rocket experiment for the first photon-counting X-ray imaging and spectroscopy of the Sun.