The first observation of the decay \(\Xi^-_\mathrm{b}\) \(\to\) \(\psi\)(2S)\(\Xi^-\) and measurement of the branching ratio of \(\Xi^-_\mathrm{b}\) \(\to\) \(\psi\)(2S)\(\Xi^-\) to \(\Xi^-_\mathrm{b}\) \(\to\) J/\(\psi\)$\Xi^-\( are presented. The J/\)\psi\( and \)\psi\((2S) mesons are reconstructed using their dimuon decay modes. The results are based on proton-proton colliding beam data from the LHC collected by the CMS experiment at \)\sqrt{s}\( = 13 TeV in 2016-2018, corresponding to an integrated luminosity of 140 fb\)^{-1}\(. The branching fraction ratio is measured to be \)\mathcal{B}\((\)\Xi^-_\mathrm{b}\( \)\to\( \)\psi\((2S)\)\Xi^-\()/\)\mathcal{B}\((\)\Xi^-_\mathrm{b}\( \)\to\( J/\)\psi\(\)\Xi^-\() = 0.84\)^{+0.21}_{-0.19}\( (stat) \)\pm\( 0.10 (syst) \)\pm\( 0.02 (\)\mathcal{B}\(), where the last uncertainty comes from the uncertainties in the branching fractions of the charmonium states. New measurements of the \)\Xi_\mathrm{b}^{\ast{}0}\( baryon mass and natural width are also presented, using the \)\Xi_\mathrm{b}^-\pi^+\( final state, where the \)\Xi^-_\mathrm{b}\( baryon is reconstructed through the decays J/\)\psi \Xi^-\(, \)\psi\((2S)\)\Xi^-\(, J/\)\psi \Lambda\(K\)^-\(, and J/\)\psi \Sigma^0\(K\)^-\(. Finally, the fraction of the \)\Xi^-_\mathrm{b}\( baryons produced from \)\Xi_\mathrm{b}^{\ast{}0}$ decays is determined.