  The latest source catalog of the Fermi-LAT telescope contains more than 7000
$\gamma$-ray sources at GeV energies, with the two dominant source classes
thought to be blazars and rotation-powered pulsars. Our target is the
identification of possible (young and recycled) pulsar candidates in the sample
of 2600 unassociated $\gamma$-ray sources, via their characteristic X-ray and
$\gamma$-ray emission. To achieve this, we cross-match the Fermi-LAT catalog
with the catalog of X-ray sources in the western Galactic hemisphere from the
first four all-sky surveys of SRG/eROSITA. We complement this by identifying
X-ray counterparts of known pulsars detected at $\gamma$-ray and radio energies
in the eROSITA data. We use a Bayesian cross-matching scheme to construct a
probabilistic catalog of possible pulsar-type X-ray counterparts to Fermi-LAT
sources. Our method combines the overlap of X-ray and $\gamma$-ray source
positions with a probabilistic classification (into pulsar and blazar
candidates) of each source based on its $\gamma$-ray properties and a
prediction on the X-ray flux of pulsar- or blazar-type counterparts. We provide
a catalog of our prior $\gamma$-ray-based classifications of all 2600
unassociated sources in the Fermi-LAT catalog, with around equal numbers of
pulsar and blazar candidates. Our final list of candidate X-ray pulsar
counterparts, cleaned for spurious detections and sources with obvious
non-pulsar counterparts, contains around 900 X-ray sources. We predict between
30 and 40 new pulsars among our top 200 candidates, with around equal expected
numbers of young and recycled pulsars. This candidate list may serve as input
to future follow-up campaigns, looking directly for pulsations or for the
orbital modulation of possible binary companions. We furthermore detect the
X-ray counterparts of 15 known rotation-powered pulsars, which were not seen in
X-rays before.
