  Planetary nebulae in Galactic open star clusters are rare objects; only three
are known to date. They are of particular interest because their distance can
be determined with high accuracy, allowing one to characterize the physical
properties of the planetary nebula and its ionizing central star with high
confidence. Here we present the first quantitative spectroscopic analysis of a
central star in an open cluster, namely the faint nucleus of IPHASX
J055226.2$+$323724 in M37. This cluster contains 14 confirmed white dwarf
members, which were previously used to study the initial-to-final-mass relation
of white dwarfs, and six additional white dwarf candidates. We performed an
atmosphere modeling of spectra taken with the 10m Gran Telescopio Canarias. The
central star is a hot hydrogen-deficient white dwarf with an effective
temperature of 90,000 K and spectral type PG1159 (helium- and carbon-rich). We
know it is about to transform into a helium-rich DO white dwarf because the
relatively low atmospheric carbon abundance indicates ongoing gravitational
settling of heavy elements. The star belongs to a group of hot white dwarfs
that exhibit ultrahigh-excitation spectral lines possibly emerging from
shock-heated material in a magnetosphere. We find a relatively high stellar
mass of $M= 0.85^{+0.13}_{-0.14}$ M$_\odot$. This young white dwarf is
important for the semi-empirical initial-final mass relation because any
uncertainty related to white-dwarf cooling theory is insignificant with respect
to the pre-white-dwarf timescale. Its post-asymptotic-giant-branch age of
$170,000-480,000$ yr suggests that the extended planetary nebula is
extraordinarily old. We also performed a spectroscopic analysis of the six
other white dwarf candidates of M37, confirming one as a cluster member.
