We analyze new SCUBA-2 submillimeter and archival SPIRE far-infrared imaging of a z = 1.62 cluster, Cl 0218.3-0510, which lies in the UKIRT Infrared Deep Sky Survey/Ultra-Deep Survey field of the SCUBA-2 Cosmology Legacy Survey. Combining these tracers of obscured star-formation activity with the extensive photometric and spectroscopic information available for this field, we identify 31 far-infrared/submillimeter-detected probable cluster members with bolometric luminosities gsim1012 L ⊙ and show that by virtue of their dust content and activity, these represent some of the reddest and brightest galaxies in this structure. We exploit ALMA submillimeter continuum observations, which cover one of these sources, to confirm the identification of a SCUBA-2-detected ultraluminous star-forming galaxy in this structure. Integrating the total star-formation activity in the central region of the structure, we estimate that it is an order of magnitude higher (in a mass-normalized sense) than clusters at z ~ 0.5-1. However, we also find that the most active cluster members do not reside in the densest regions of the structure, which instead host a population of passive and massive, red galaxies. We suggest that while the passive and active populations have comparable near-infrared luminosities at z = 1.6, MH ~ -23, the subsequent stronger fading of the more active galaxies means that they will evolve into passive systems at the present day that are less luminous than the descendants of those galaxies that were already passive at z ~ 1.6 (MH ~ -20.5 and MH ~ -21.5, respectively, at z ~ 0). We conclude that the massive galaxy population in the dense cores of present-day clusters were already in place at z = 1.6 and that in Cl 0218.3-0510 we are seeing continuing infall of less extreme, but still ultraluminous, star-forming galaxies onto a pre-existing structure.