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Swift J1858.6-0814 was discovered by Swift-BAT on October 25, 2018. Here we report on the first follow-up NuSTAR observation of the source, which shows variability spanning two orders of magnitude in count rate on timescales of ~10-100 s. The power-spectrum of the source does not show any quasi-periodic oscillations or periodicity, but has a large fractional rms amplitude of 147%$\pm3$%, exhibiting a number of large flares throughout the observation. The hardness ratio (defined as $R_{10-79 \rm keV}/R_{3-10 \rm keV}$) of the flares tends to be soft, while the source spans a range of hardness ratios during non-flaring periods. The X-ray spectrum of the source shows strong reflection features, which become more narrow and peaked during the non-flaring intervals. We fit an absorbed relativistic reflection model to the source spectra to place physical constraints on the system. Most notably, we find that the source exhibits a large and varying intrinsic absorbing column density ($N_{\rm H}=1.4-4.2\times10^{23}$ cm$^{-2}$). This large intrinsic absorption is further supported by the energy spectra extracted from two flares observed simultaneously by NuSTAR and NICER. We find that the inner accretion disk of the source has a low inclination, $i<29^{\circ}$ ( 3$σ$ upper-limit), while the iron abundance in the disk is close to solar, $A_{\rm Fe}=1.0\pm0.3$. We set a 90% confidence upper limit on the inner radius of the accretion disk of $r_{\rm in}<8 \ r_{\rm ISCO}$, and, by fixing $r_{\rm in}$ to be at $r_{\rm ISCO}$, a 90% confidence lower-limit on the spin of the black hole of $a^{*}>0.0$. Lastly, we compare the properties of Swift J1858.6$-$0814 to those of V404 Cygni and V4641 Sgr, which both show rapid flaring and a strong and variable absorption.