By implementing a model of primordial dust emission, we predict dust-continuum fluxes for massive galaxy sources similar to those recently detected by James Webb Space Telescope (JWST) at z ≳ 7. Current upper flux limits, obtained with Atacama Large Millimeter/submillimeter Array (ALMA) for some of these sources, can constrain the gas metallicity and dust fraction of the first galaxies. Encouragingly, if assuming expected properties for typical first galaxies (i.e., dust-to-metal mass ratio: D/M = 5 × 10−3, gas metallicity: Z
g = 5 × 10−3
Z
☉, star formation efficiency: η = 0.01), model far-infrared (FIR) fluxes are consistent with current upper flux limits inferred from ALMA bands 6 and 7 (≲104 nJy). Such low D/M values and metallicities are in agreement with some scenarios proposed in the literature to explain the nondetection of the FIR dust continuum for high-z JWST galaxy candidates. On the other hand, higher values of model parameters D/M (≳0.06) and Z
g (≳5 × 10−2
Z
☉) are ruled out by observational data, unless a higher η is assumed. According to our findings, ALMA multiband observations could constrain the dust chemistry and dust grain size distribution in the early universe. In this context, future observational challenges would involve not only reaching higher FIR sensitivities, but also increasing the wavelength coverage by exploring distinct ALMA bands.