The $s$-process in massive stars produces the weak component of the $s$-process (nuclei up to $A \sim 90$), in amounts that match solar abundances. For heavier isotopes, such as barium, production through neutron capture is significantly enhanced in very metal-poor stars with fast rotation. However, detailed theoretical predictions for the resulting final $s$-process abundances have important uncertainties caused both by the underlying uncertainties in the nuclear physics (principally neutron capture reaction and $\beta$-decay rates) as well as by the stellar evolution modeling. In this work, we investigated the impact of nuclear-physics uncertainties relevant to the $s$-process in massive stars. Using a Monte-Carlo based approach, we performed extensive nuclear reaction network calculations that include newly evaluated upper and lower limits for the individual temperature dependent reaction rates. We found that most of the uncertainty in the final abundances is caused by uncertainties in the neutron capture rates, while $\beta$-decay rate uncertainties affect only a few nuclei near $s$-process branchings. The $s$-process in rotating metal-poor stars shows quantitatively different uncertainties and key reactions, although the qualitative characteristics are similar. We confirmed that our results do not significantly change at different metallicities for fast rotating massive stars in the very low metallicity regime. We highlight which of the identified key reactions are realistic candidates for improved measurement by future experiments.