Abstract: The MoEDAL's Apparatus for Penetrating Particles (MAPP) Experiment is designed to expand the search for new physics at the LHC, significantly extending the physics program of the baseline MoEDAL Experiment. The Phase-1 MAPP detector (MAPP-1) is currently undergoing installation at the LHC's UA83 gallery adjacent to the LHCb/MoEDAL region at Interaction Point 8 and will begin data-taking in early 2024. The focus of the MAPP experiment is on the quest for new feebly interacting particles – avatars of new physics with extremely small Standard Model couplings, such as minicharged particles (mCPs). In this study, we present the results of a comprehensive analysis of MAPP-1's sensitivity to mCPs arising in the canonical model involving the kinetic mixing of a massless dark U(1) gauge field with the Standard Model hypercharge gauge field. We focus on several dominant production mechanisms of mCPs at the LHC across the mass-mixing parameter space of interest to MAPP: Drell-Yan pair production, direct decays of heavy quarkonia and light vector mesons, and single Dalitz decays of pseudoscalar mesons. The 95% confidence level background-free sensitivity of MAPP-1 for mCPs produced at the LHC's Run 3 and the HL-LHC through these mechanisms, along with projected constraints on the minicharged strongly interacting dark matter window, are reported. Our results indicate that MAPP-1 exhibits sensitivity to sizable regions of unconstrained parameter space and can probe effective charges as low as 8 x 10 -4 e and 6 x 10 -4 e for Run 3 and the HL-LHC, respectively.