The 233U nucleus plays a key role as the fissile isotope in the Th-U fuel cycle. Consequently the accurate knowledge of its fission and capture cross sections is essential for a potential reactor design. As highlighted by the NEA High Priority Request List, the 233U(n,g) is of high importance for the production and destruction of 233U and is required to facilitate defining the reprocessing scheme for a molten salt reactor. Due to the 233U(n,f) reaction's inherent gamma-ray background the measurement of 233U(n,g) requires an efficient background discrimination. This can be achieved by employing a fission and a gamma-ray detector in anti-coincidence. For this purpose a pocket-sized fast fission chamber was designed and used together with the n_TOF Total Absorption Calorimeter 4$\pi$ gamma-ray detector at the n_TOF EAR1 facility at CERN. A brief description of the experimental setup is given and the fission detector and coupled detectors' performance will be presented. A description of the analysis methodology and procedure discussing relevant sources of background for the measurement will be given. The setup's capability to identify the fission gamma-rays for background estimation in turn allows us to extract information on the average energy and multiplicity of the prompt fission gamma-rays which are compared with other recent measurements and theoretical models. Furthermore, a careful analysis of the coincidence spectra enables to study the half-lives of 233U isomeric fission states. A good agreement with simulated gamma-ray cascades of the captured nucleus allows us to calculate the experimental detection efficiencies and finally the results on the 233U $\alpha$-ratio and 233U(n,g) will be shown and compared to existing data sets and evaluations.