We present the first spatially resolved observations of molecular gas in a sample of cluster galaxies beyond z > 0.1. Using ALMA, we detect CO (2-1) in eight z ∼ 1.6 cluster galaxies, all within a single 70″ primary beam. The cluster, SpARCS-J0225, was discovered by the Spitzer Adaptation of the Red-sequence Cluster Survey, and is replete with gas-rich galaxies in close proximity, thus affording an efficient multiplexing strategy to amass the first sample of resolved CO in distant clusters. Mapping out the kinematic structure and morphology of molecular gas on ∼3.5 kpc scales reveals rotating gas disks in the majority of the galaxies, and some kinematic peculiarities, including a central gas void, a merger, and one-sided gas tails. We find that the extent of the molecular gas is slightly smaller than that of the optical HST stellar component; this is even more pronounced in low-redshift Virgo cluster galaxies. However, limited by small sample sizes of spatially resolved CO, we are unable to differentiate the distribution of stellar-to-gas radii between cluster and field environments at high redshift. Thus, at first glance, while the cluster galaxies generally look like galaxies infalling from the field, with typical main-sequence star formation rates and massive molecular gas reservoirs situated in rotating disks, they have potentially remarkable attributes, including elevated gas fractions, slightly smaller CO disks, and asymmetric gas tails. Taken in tandem, these signatures are tentative evidence for gas stripping in the z ∼ 1.6 cluster, though verification of these trends will require larger samples.