There is a global trend to increase the light yield of CsI scintillators used in neutrino and dark matter detection by operating undoped crystals at cryogenic temperatures. However, high light yield alone is not sufficient to guarantee a low-energy threshold. The response of undoped crystals to nuclear recoils at cryogenic temperatures is equally important. A liquid nitrogen-based cryostat was developed to measure the nuclear quenching factor of a small undoped CsI crystal using monoenergetic neutron beams at the Triangle Universities Nuclear Laboratory (TUNL). To avoid neutron scattering in high-Z materials, these materials were intentionally reduced around the crystal. The structure and performance of the cryostat are described in detail. Using this cryostat, a light yield of 33.4 ± 1.7 photoelectrons per keV electron-equivalent (PE/keV ee ) was observed at 5.9 keV ee , enabling the measurement of nuclear quenching factors at very low energies. The results of the quenching factor measurement will be reported in a subsequent paper.Non-negligible overshoot was observed in the tails of the observed light pulses. The origin of this issue and the correction procedure are described in detail. This information may be useful for others who encounter similar technical challenges.