The bright Type II-plateau supernova (SN) 2004dj occurred within the young, massive stellar cluster Sandage-96 in a spiral arm of NGC 2403. New multiwavelength observations obtained with several ground-based and space-based telescopes were combined to study the radiation from Sandage-96 after SN 2004dj faded away. Sandage-96 started to dominate the flux in the optical bands starting from 2006 September (~800 days after explosion). The optical fluxes are equal to the pre-explosion ones within the observational uncertainties. An optical Keck spectrum obtained ~900 days after explosion shows the dominant blue continuum from the cluster stars shortward of 6000 Å as well as strong SN nebular emission lines redward. The integrated spectral energy distribution (SED) of the cluster has been extended into the ultraviolet region by archival XMM-Newton and new Swift observations, and compared with theoretical models. The outer parts of the cluster have been resolved by the Hubble Space Telescope, allowing the construction of a color-magnitude diagram (CMD). The fitting of the cluster SED with theoretical isochrones results in cluster ages distributed between 10 and 40 Myr, depending on the assumed metallicity and the theoretical model family. The isochrone fitting of the CMDs indicates that the resolved part of the cluster consists of stars having a bimodal age distribution: a younger population at ~10-16 Myr and an older one at ~32-100 Myr. The older population has an age distribution similar to that of the other nearby field stars. This may be explained with the hypothesis that the outskirts of Sandage-96 are contaminated by stars captured from the field during cluster formation. The young age of Sandage-96 and the comparison of its pre and postexplosion SEDs suggest 12 lsim M prog lsim 20 M sun as the most probable mass range for the progenitor of SN 2004dj. This is consistent with, but perhaps slightly higher than, most of the other Type II-plateau SN progenitor masses determined so far.