The NUCLEUS experiment aims for the detection of coherent elastic neutrino-nucleus scattering at a nuclear power reactor with gram-scale, ultra-low-threshold cryogenic detectors. This technology leads to a miniaturization of neutrino detectors and allows to probe physics beyond the Standard Model of particle physics. A 0.5 g NUCLEUS prototype detector, operated above ground in 2017, reached an energy threshold for nuclear recoils of below 20 eV. This sensitivity is achieved with tungsten transition edge sensors which are operating at temperatures of 15 mK and are mainly sensitive to non-thermal phonons. These small recoil energies become accessible for the first time with this technology, which allows collecting large-statistics neutrino event samples with a moderate detector mass. A first-phase cryogenic detector array with a total mass of 10 g enables a 5-sigma observation of coherent scattering within several weeks. We identified a suitable experimental site at the Chooz Nuclear Power Plant and performed muon and neutron background measurements there. The operation of a NUCLEUS cryogenic detector array at such a site requires highly efficient background suppression. NUCLEUS plans to use an innovative technique consisting of separate cryogenic anticoincidence detectors against surface backgrounds and penetrating (gamma, neutron) radiation. We present first results from prototypes of these veto detectors and their operation in coincidence with a NUCLEUS target detector.