Next generation observatories will enable us to study the first billion years of our Universe in unprecedented detail. Foremost among these are 21-cm interferometry with the HERA and the SKA, and high-$z$ galaxy observations with the James Webb Space Telescope (JWST). Taking a basic galaxy model, in which we allow the star formation rates and ionizing escape fractions to have a power-law dependence on halo mass with an exponential turnover below some threshold, we quantify how observations from these instruments can be used to constrain the astrophysics of high-$z$ galaxies. For this purpose, we generate mock JWST LFs, based on two different hydrodynamical cosmological simulations; these have intrinsic luminosity functions (LFs) which turn over at different scales and yet are fully consistent with present-day observations. We also generate mock 21-cm power spectrum observations, using 1000h observations with SKA1 and a moderate foreground model. Using only JWST data, we predict up to a factor of 2-3 improvement (compared with HST) in the fractional uncertainty of the star formation rate to halo mass relation and the scales at which the LFs peak (i.e. turnover). Most parameters regulating the UV galaxy properties can be constrained at the level of $sim 10$% or better, if either (i) we are able to better characterize systematic lensing uncertainties than currently possible; or (ii) the intrinsic LFs peak at magnitudes brighter than $M_{rm UV} lesssim -13$. Otherwise, improvement over HST-based inference is modest. When combining with upcoming 21-cm observations, we are able to significantly mitigate degeneracies, and constrain all of our astrophysical parameters, even for our most pessimistic assumptions about upcoming JWST LFs. The 21-cm observations also result in an order of magnitude improvement in constraints on the EoR history.
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