The massive binary system Eta Carinae is characterized by intense colliding winds that form shocks and emit X-rays. The system is highly eccentric (e ≃ 0.9), resulting in modulated X-ray emission during its 5.54 yr orbit. The X-ray flux increases in the months prior to periastron passage, exhibiting strong flares, then rapidly declines to a flat minimum lasting a few weeks, followed by a gradual recovery. We present the Neutron Star Interior Composition Explorer telescope spectra obtained before, during, and after the 2020 X-ray minimum, and perform spectral analysis to establish the temporal behavior of the X-ray flux and X-ray-absorbing column density (N H(t)) for the 2-10 keV and 5-10 keV energy ranges. The latter range is dominated by the stellar wind-collision region and, therefore, these spectral parameters - in particular, N H(t) - serve as potentially stringent constraints on the binary orientation. We compare the observed N H(t) results to the behavior predicted by a simple geometrical model in an attempt to ascertain which star is closer to us at periastron: the more massive primary (ω ≃ 240 -270 ) or the secondary (ω ≃ 90 ). We find that the variations in column density, both far from periastron and around periastron passage, support the latter configuration (ω ≃ 90 ). The 2020 X-ray minimum showed the fastest recovery among the last five minima, providing additional evidence for a recent weakening of the primary star's wind.