  The bright star $\pi$ Men was chosen as the first target for a radial
velocity follow-up to test the performance of ESPRESSO, the new high-resolution
spectrograph at the ESO's Very-Large Telescope (VLT). The star hosts a
multi-planet system (a transiting 4 M$_\oplus$ planet at $\sim$0.07 au, and a
sub-stellar companion on a $\sim$2100-day eccentric orbit) which is
particularly appealing for a precise multi-technique characterization. With the
new ESPRESSO observations, that cover a time span of 200 days, we aim to
improve the precision and accuracy of the planet parameters and search for
additional low-mass companions. We also take advantage of new photometric
transits of $\pi$ Men c observed by TESS over a time span that overlaps with
that of the ESPRESSO follow-up campaign. We analyse the enlarged spectroscopic
and photometric datasets and compare the results to those in the literature. We
further characterize the system by means of absolute astrometry with Hipparcos
and Gaia. We used the spectra of ESPRESSO for an independent determination of
the stellar fundamental parameters. We present a precise characterization of
the planetary system around $\pi$ Men. The ESPRESSO radial velocities alone
(with typical uncertainty of 10 cm/s) allow for a precise retrieval of the
Doppler signal induced by $\pi$ Men c. The residuals show an RMS of 1.2 m/s,
and we can exclude companions with a minimum mass less than $\sim$2 M$_\oplus$
within the orbit of $\pi$ Men c). We improve the ephemeris of $\pi$ Men c using
18 additional TESS transits, and in combination with the astrometric
measurements, we determine the inclination of the orbital plane of $\pi$ Men b
with high precision ($i_{b}=45.8^{+1.4}_{-1.1}$ deg). This leads to the precise
measurement of its absolute mass $m_{b}=14.1^{+0.5}_{-0.4}$ M$_{Jup}$, and
shows that the planetary orbital planes are highly misaligned.
