  Context: Observations of the hot gas in distant clusters of galaxies, though
challenging, are key to understand the role of intense galaxy activity,
super-massive black hole feedback and chemical enrichment in the process of
massive halos assembly. Aims: We assess the feasibility to retrieve, using
X-ray hyperspectral data only, the thermodymamical hot gas properties and
chemical abundances of a $z=2$ galaxy cluster of mass M500=7 x $10^{13}
M_{\odot}$, extracted from the Hydrangea hydrodynamical simulation. Methods: We
create mock X-ray observations of the future X-ray Integral Field Unit (X-IFU)
onboard the Athena mission. By forward-modeling the measured 0.4-1 keV surface
brightness, the projected gas temperature and abundance profiles, we
reconstruct the three-dimensional distribution for the gas density, pressure,
temperature and entropy. Results: Thanks to its large field-of-view, high
throughput and exquisite spectral resolution, one X-IFU exposure lasting 100ks
enables reconstructing density and pressure profiles with 20% precision out to
a characteristic radius of R500, accounting for each quantity's intrinsic
dispersion in the Hydrangea simulations. Reconstruction of abundance profiles
requires both higher signal-to-noise ratios and specific binning schemes. We
assess the enhancement brought by longer exposures and by observing the same
object at later evolutionary stages ($z=1-1.5$). Conclusions: Our analysis
highlights the importance of scatter in the radially binned gas properties,
which induces significant effects on the observed projected quantities. The
fidelity of the reconstruction of gas profiles is sensitive to the degree of
gas components mixing along the line-of-sight. Future analyses should aim at
involving dedicated hyper-spectral models and fitting methods that are able to
grasp the complexity of such three-dimensional, multi-phase, diffuse gas
structures.
