We report high-precision X-ray monitoring observations in the 0.4-10keV band of the luminous, long-period colliding wind binary Eta Carinae, up to and through its most recent X-ray minimum/periastron passage in 2020 February. Eta Carinae reached its observed maximum X-ray flux on 2020 January 7, at a flux level of 3.30x10^-10^ergs/s/cm2, followed by a rapid plunge to its observed minimum flux, 0.03x10^-10^ergs/s/cm2, near 2020 February 17. The NICER observations show an X-ray recovery from the minimum of only ~16 days, the shortest X-ray minimum observed so far. We provide new constraints for the "deep" and "shallow" minimum intervals. Variations in the characteristic X-ray temperatures of the hottest observed X-ray emission indicate that the apex of the wind-wind "bow shock" enters the companion's wind acceleration zone about 81 days before the start of the X-ray minimum. There is a steplike increase in column density just before the X-ray minimum, probably associated with the presence of dense clumps near the shock apex. During the recovery and after, the column density shows a smooth decline, which agrees with previous NH measurements made by Swift at the same orbital phase, indicating that the changes in the mass-loss rate are only a few percent over the two cycles. Finally, we use the variations in the X-ray flux of the outer ejecta seen by NICER to derive a kinetic X-ray luminosity of the ejecta of ~10^41^ergs/s near the time of the "Great Eruption."