We present ALMA CO (2-1) detections in 11 gas-rich cluster galaxies at z similar to 1.6, constituting the largest sample of molecular gas measurements in z > 1.5 clusters to date. The observations span three galaxy clusters, derived from the Spitzer Adaptation of the Red-sequence Cluster Survey. We augment the > 5 sigma detections of the CO (2-1) fluxes with multi-band photometry, yielding stellar masses and infrared-derived star formation rates, to place some of the first constraints on molecular gas properties in z similar to 1.6 cluster environments. We measure sizable gas reservoirs of 0.5-2 x 10(11) M circle dot in these objects, with high gas fractions (f(gas)) and long depletion timescales (tau), averaging 62% and 1.4 Gyr, respectively. We compare our cluster galaxies to the scaling relations of the coeval field, in the context of how gas fractions and depletion timescales vary with respect to the star-forming main sequence. We find that our cluster galaxies lie systematically off the field scaling relations at z = 1.6 toward enhanced gas fractions, at a level of similar to 4 sigma, but have consistent depletion timescales. Exploiting CO detections in lower-redshift clusters from the literature, we investigate the evolution of the gas fraction in cluster galaxies, finding it to mimic the strong rise with redshift in the field. We emphasize the utility of detecting abundant gas-rich galaxies in high-redshift clusters, deeming them as crucial laboratories for future statistical studies.