The experimental concept of a search for a long-range coupling between rubidium (Rb) nuclear spins and the mass of the Earth is described. The experiment is based on simultaneous measurement of the spin precession frequencies for overlapping ensembles of 85Rb and 87Rb atoms contained within an evacuated, antirelaxation-coated vapor cell. Rubidium atoms are spin-polarized in the presence of an applied magnetic field by synchronous optical pumping with circularly polarized laser light. Spin precession is probed by measuring optical rotation of far-off-resonant, linearly polarized laser light. Simultaneous measurement of 85Rb and 87Rb spin precession frequencies enables suppression of magnetic-field-related systematic effects. The nuclear structure of the Rb isotopes makes the experiment particularly sensitive to anomalous spin-dependent interactions of the proton. Experimental sensitivity and a variety of systematic effects are discussed, and initial data are presented.