We present Atacama Large Millimeter/submillimeter Array observations of the 870 μm continuum and CO(4-3) line emission in the core of the galaxy cluster Cl J1449+0856 at z = 2, a near-IR-selected, X-ray-detected system in the mass range of typical progenitors of today's massive clusters. The 870 μm map reveals six F 870μm > 0.5 mJy sources spread over an area of 0.07 arcmin2, giving an overdensity of a factor of ∼10 (6) with respect to blank-field counts down to F 870μm > 1 mJy (>0.5 mJy). On the other hand, deep CO(4-3) follow-up confirms membership of three of these sources but suggests that the remaining three, including the brightest 870 μm sources in the field (F 870μm ≳2 mJy), are likely interlopers. The measurement of 870 μm continuum and CO(4-3) line fluxes at the positions of previously known cluster members provides a deep probe of dusty star formation occurring in the core of this high-redshift structure, adding up to a total star formation rate of ∼700 ±100 M o yr-1 and yielding an integrated star formation rate density of ∼104 M o yr-1 Mpc-3, five orders of magnitude larger than in the field at the same epoch, due to the concentration of star-forming galaxies in the small volume of the dense cluster core. The combination of these observations with previously available Hubble Space Telescope imaging highlights the presence in this same volume of a population of galaxies with already suppressed star formation. This diverse composition of galaxy populations in Cl J1449+0856 is especially highlighted at the very cluster center, where a complex assembly of quiescent and star-forming sources is likely forming the future brightest cluster galaxy.