We report results of XMM-Newton observations of a γ-ray pulsar J0633+0632 and its wind nebula. We reveal, for the first time, pulsations of the pulsar X-ray emission with a single sinusoidal pulse-profile and a pulsed fraction of 23 ± 6 per cent in the 0.3-2 keV band. We confirm previous Chandra findings that the pulsar X-ray spectrum consists of thermal and non-thermal components. However, we do not find the absorption feature that was previously detected at about 0.8 keV. Thanks to the greater sensitivity of XMM-Newton, we get stronger constraints on spectral model parameters compared to previous studies. The thermal component can be equally well described by either blackbody or neutron star atmosphere models, implying that this emission is coming from either hot pulsar polar caps with a temperature of about 120 eV or from the colder bulk of the neutron star surface with a temperature of about 50 eV. In the latter case, the pulsar appears to be one of the coolest among other neutron stars of similar ages with estimated surface temperatures. We discuss cooling scenarios relevant to this neutron star. Using an interstellar absorption-distance relation, we also constrain the distance to the pulsar to the range of 0.7-2 kpc. Besides the pulsar and its compact nebula, we detect regions of weak large-scale diffuse non-thermal emission in the pulsar field and discuss their possible nature.