  By implementing a model of primordial dust emission, we predict
dust-continuum fluxes for massive galaxy sources similar to those recently
detected by JWST at $z \gtrsim 7$. Current upper flux limits, obtained with
ALMA for some of these sources, can constrain 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: $D/M = 5 \times
10^{-3}$, gas metallicity: $Z_{\rm g} = 5 \times 10^{-3}~Z_{\odot}$, star
formation efficiency: $\eta = 0.01$), model far-infrared (FIR) fluxes are
consistent with current upper flux limits inferred from ALMA bands 6 and 7
($\lesssim 10^4$ nJy). Such low $D/M$ values and metallicities are in agreement
with some scenarios proposed in the literature to explain the non-detection of
the FIR dust continuum for high-$z$ JWST galaxy candidates. On the other hand,
higher values of model parameters $D/M$ ($\gtrsim 0.06$) and $Z_{\rm g}$
($\gtrsim 5 \times 10^{-2}~Z_{\odot}$) are ruled out by observational data,
unless a higher $\eta$ is assumed. According to our findings, ALMA multi-band
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.
