A real-time Data Acquisition (DAQ) system for the CULTASK axion haloscope experiment was constructed and tested. The CULTASK is an experiment to search for cosmic axions using resonant cavities, to detect photons from axion conversion through the inverse Primakoff effect in a few GHz frequency range in a very high magnetic field and at an ultra low temperature. The constructed DAQ system utilizes a Field Programmable Gate Array (FPGA) for data processing and Fast Fourier Transformation. This design along with a custom Ethernet packet designed for real-time data transfer enables 100% DAQ efficiency, which is the key feature compared with a commercial spectrum analyzer. This DAQ system is optimally designed for RF signal detection in the axion experiment, with 100 Hz frequency resolution and 500 kHz analysis window. The noise level of the DAQ system averaged over 100,000 measurements is around -111.7 dBm. From a pseudo-data analysis, an improvement of the signal-to-noise ratio due to repeating and averaging the measurements using this real-time DAQ system was confirmed.