The study of short-term variability properties in AGN jets has the potential to shed light on their particle acceleration and emission mechanisms. We report results from a four-day coordinated multi-wavelength campaign on the highly-peaked blazar (HBL) Mkn 421 in January 2019. We obtained X-ray data from AstroSAT, BVRI photometry with the Whole Earth Blazar Telescope (WEBT), and TeV data from FACT to explore short-term multi-wavelength variability in this HBL. The X-ray continuum is rapidly variable on time-scales of tens of ks. Fractional variability amplitude increases with energy across the synchrotron hump, consistent with previous studies; we interpret this observation in the context of a model with multiple cells whose emission spectra contain cutoffs that follow a power-law distribution. We also performed time-averaged and time-resolved (time-scales of 6 ks) spectral fits; a broken power-law model fits all spectra well; time-resolved spectral fitting reveals the usual hardening when brightening behaviour. Intra-X-ray cross correlations yield evidence for the 0.6-0.8 keV band to likely lead the other bands by an average of 4.6 ± 2.6 ks, but only during the first half of the observation. The source displayed minimal night-to-night variability at all wavebands thus precluding significant interband correlations during our campaign. The broadband SED is modeled well with a standard one-zone leptonic model, yielding jet parameters consistent with those obtained from previous SEDs of this source.