We employ three-dimensional state-of-the-art magnetohydrodynamic models of the early solar wind and heliosphere and a two-dimensional model for cosmic-ray transport to investigate the cosmic-ray spectrum and flux near the Archean Earth. We assess how sensitive the cosmic-ray spectrum is to changes in the sunspot placement and magnetic field strength, the large-scale dipole magnetic field strength, the wind ram pressure, and the Sun's rotation period. Overall, our results confirm earlier work that suggested the Archean Earth would have experienced a greatly reduced cosmic-ray flux than is the case today. The cosmic-ray reduction for the early Sun is mainly due to the shorter solar rotation period and tighter winding of the Parker spiral, and to the different surface distribution of the more active solar magnetic field. These effects lead to a global reduction of the cosmic-ray flux at 1 AU by up to two orders of magnitude or more. Variations in the sunspot magnetic field have more effect on the flux than variations in the dipole field component. The wind ram pressure affects the cosmic-ray flux through its influence on the size of the heliosphere via the pressure balance with the ambient interstellar medium. Variations in the interstellar medium pressure experienced by the solar system in orbit through the Galaxy could lead to order of magnitude changes in the cosmic-ray flux at Earth on timescales of a few million years.