A new magneto-optical trap-target recoil ion momentum spectroscopy apparatus has been built and tested at the LPC-CAEN. Dedicated to ion-atom collisions studies and excited fraction measurements, the setup combines a projectile ion beam line, a target of cold rubidium atoms provided by a magneto-optical trap (MOT), and a recoil ion momentum spectrometer. In a test experiment using a beam of Na(+) projectiles, we demonstrate its capability to measure, with a very high signal over background ratio, fully differential cross sections in scattering angle, initial state, and final state of the system. We detail, in this work, features that had not been described previously in the literature: an extraction of the recoil ions transverse to the ion beam axis, and a fast switch for the MOT magnetic field. Advantages of transverse versus longitudinal extraction are discussed, and future possibilities for the setup are presented.