The Cherenkov Telescope Array (CTA) is the next generation ground-based observatory for $\gamma$-ray astronomy at energies above 30 GeV. Thanks to its unique capabilities, CTA observations will address a plethora of open questions in astrophysics, ranging from the origin of cosmic messengers to the exploration of the frontiers of physics. In this note, we present a comprehensive sensitivity study to assess the potential of CTA to measure the $\gamma$-ray absorption on the extragalactic background light (EBL), to constrain or detect intergalactic magnetic fields (IGMFs), and probe physics beyond the standard model such as axion-like particles (ALPs) and Lorentz invariance violation (LIV), which could modify the $\gamma$-ray spectra features expected from EBL absorption. Our results suggest that CTA will have unprecedented sensitivity to detect IGMF signatures and will probe so-far unexplored regions of the LIV and ALP parameter space. Furthermore, an indirect measurement of the EBL and of its evolution will be performed with unrivaled precision.