Magneto-Raman spectroscopy is applied to study spin-phonon coupling in the layered two-dimensional (2D) van der Waals antiferromagnet CrSBr. We report on the effects of temperature and external magnetic field on Raman-active phonons of $A_g$ symmetry in bulk and one-to-six-layer forms of CrSBr that are reflected by the Raman spectral patterns measured at different configurations of circularly polarized laser beam. The results of experimental investigations reveal that spin-spin and spin-phonon interactions play a significant role below the Néel temperature and are notably stronger in CrSBr monolayer than in bulk material. Spin-phonon coupling leads to the emergence of \textit{ new} phonon peaks at very low temperatures. Such effect is solely observed in one-to-six-layer CrSBr and is absent in its bulk form. Our experimental research is accompanied by \textit{ab initio} simulations of the Raman spectra of bulk and monolayer CrSBr that uncover correlations between intensities of Raman-active phonons and the arrangement of spin magnetic moments on Cr atoms. Comparative analysis of simulated and experimental Raman spectra suggests the most favored spin orientation in the CrSBr antiferromagnetic system.