We present the results of a ~60-hr observational campaign with ALMA targeting a spectroscopically confirmed and lensed sub-$L^\star$ galaxy at z=6.07, identified during the ALMA Lensing Cluster Survey (ALCS). We sample the dust continuum emission from rest frame 90 to 370 $\mu$m at six different frequencies and set constraining upper limits on the molecular gas line emission and content via CO(7-6) and [CI](2-1) for two lensed images with $\mu\gtrsim20$. Complementing these sub-mm observations with deep optical and near-IR photometry and spectroscopy with JWST, we find this galaxy to form stars at a rate of SFR~7 Msun/yr, ~50-70% of which is obscured by dust. This is consistent with what is expected for a $M_\star$~7.5$\times10^{8}$ Msun object by extrapolating the $M_\star$-obscured SFR fraction relation at z<2.5 and with observations at 5Comment: Accepted for publication in A&A; 13 pages+appendix; supplementary figures: https://doi.org/10.5281/zenodo.10703293. See also the companion papers on arXiv: Fujimoto+2024 and Gim\'{e}nez-Arteaga+2024