We analysed Chandra observations of the bright Fermi pulsar J0633+0632 and found evidence of an absorption feature in its spectrum at 804(-26)(+42) eV (the errors are at 90% confidence) with equivalent width of 63(-36)(+47) eV. In addition, we analysed in detail the X-ray spectral continuum taking into account correlations between the interstellar absorption and the distance to the source. We confirm early findings that the spectrum contains non-thermal and thermal components. The latter is equally well described by the blackbody and magnetised atmosphere models and can be attributed to the emission from the bulk of the stellar surface in both cases. The distance to the pulsar is constrained in a range of 1-4 kpc from the spectral fits. We infer the blackbody surface temperature of 108(-14)(+22) eV, while for the atmosphere model, the temperature, as seen by a distant observer, is 53(-7)(+12) eV. In the latter case, J0633+0632 is one of the coldest middle-aged isolated neutron stars. Finally, it powers an extended pulsar wind nebula whose shape suggests a high pulsar proper motion. Looking backwards the direction of the presumed proper motion, we found a likely birthplace of the pulsar-the Rosette nebula, a 50-Myr-old active star-forming region located at about 1.5 degrees from the pulsar. If true, this constrains the distance to the pulsar in the range of 1.2-1.8 kpc.