The Monument experiment measures ordinary muon capture (OMC) on isotopes relevant for neutrinoless double-beta (0νββ) decay and nuclear astrophysics. OMC is a particularly attractive tool for improving the theoretical description of 0νββ decay. It involves similar momentum transfers and allows testing the virtual transitions involved in 0νββ decay against experimental data. During the 2021 campaign, Monument studied OMC on ⁷⁶Se and ¹³⁶Ba, the isotopes relevant for next-generation decay searches, like Legend and nEXO. The experimental setup has been designed to accurately extract the total and partial muon capture rates, which requires precise reconstruction of energies and time-dependent intensities of the OMC-related γ rays. The setup also includes a veto counter system to allow selecting a clean sample of OMC events. This work provides a detailed description of the Monument setup operated during the 2021 campaign, its two DAQ systems, calibration and analysis approaches, and summarises the achieved detector performance. Future improvements are also discussed.