  We analysed Chandra observations of the bright Fermi pulsar J0633+0632 and
found evidence of an absorption feature in its spectrum at $804^{+42}_{-26}$ eV
(the errors here and below are at 90% confidence) with equivalent width of
$63^{+47}_{-36}$ 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 by Ray et al. (2011)
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
$108^{+22}_{-14}$ eV, while for the atmosphere model, the temperature, as seen
by a distant observer, is $53^{+12}_{-7}$ eV. In the latter case J0633+0632 is
one of the coldest middle-aged isolated neutron stars with measured
temperatures. 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$.\!\!^\circ$5 from the pulsar. If true, this constrains the distance to the
pulsar in the range of 1.2--1.8 kpc.
