  Hydrogen-rich superhydrides are believed to be very promising high-T$_c$
superconductors, with experimentally observed critical temperatures near room
temperature, as shown in recently discovered lanthanide superhydrides at very
high pressures, e.g. LaH$_{10}$ at 170 GPa and CeH$_9$ at 150 GPa. With the
motivation of discovering new hydrogen-rich high-T$_c$ superconductors at
lowest possible pressure, quantitative theoretical predictions are needed. In
these promising compounds, superconductivity is mediated by the highly
energetic lattice vibrations associated with hydrogen and their interplay with
the electronic structure, requiring fine descriptions of the electronic
properties, notoriously challenging for correlated $f$ systems. In this work,
we propose a first-principles calculation platform with the inclusion of
many-body corrections to evaluate the detailed physical properties of the Ce-H
system and to understand the structure, stability and superconductivity of
CeH$_9$ at high pressure. We report how the prediction of T$_c$ is affected by
the hierarchy of many-body corrections, and obtain a compelling increase of
T$_c$ at the highest level of theory, which goes in the direction of
experimental observations. Our findings shed a significant light on the search
for superhydrides in close similarity with atomic hydrogen within a feasible
pressure range. We provide a practical platform to further investigate and
understand conventional superconductivity in hydrogen rich superhydrides.
