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We report a detailed experimental and theoretical study on the effect of hydrostatic pressure on the structural and magnetic aspects of the layered honeycomb antiferromagent $\alpha$-RuCl$_{3}$. Magnetic susceptibility measurements performed under almost ideal hydrostatic-pressure conditions yield that the phase transition to zigzag-type antiferromagnetic order at $T_N$ = 7.3 K can be rapidly suppressed to about 6.1 K. A further suppression with increasing pressure is impeded due to the occurrence of a pressure-induced structural transition at $p \geq$ 104 MPa, accompanied by a strong dimerization of Ru-Ru bonds, which gives rise to a collapse of the magnetic susceptibility. Whereas the dimerization transition is strongly first order, as reflected by large discontinuous changes in $\chi$ and pronounced hysteresis effects, the magnetic transition under varying pressure and magnetic field also reveals indications for a weakly first-order transition. We assign this observation to a strong magnetoelastic coupling in this system. Measurements of $\chi$ under varying pressure in the paramagnetic regime ($T > T_N$) and before dimerization ($p <$ 100 MPa) reveal a considerable increase of $\chi$ with pressure. These experimental observations are consistent with the results of ab-initio Density Functional Theory (DFT) calculations on the pressure-dependent structure and the corresponding pressure-dependent magnetic model. Comparative susceptibility measurements on a second crystal showing two consecutive magnetic transitions instead of one, indicating the influence of stacking faults. Using different temperature-pressure protocols the effect of these stacking faults can be temporarily overcome, transforming the magnetic state from a multiple-$T_N$ into a single-$T_N$ state.