The results of an extensive ab initio study of the cyanobutadiyne cation, initially motivated by threshold-photoelectron spectroscopy experiments [see the study by Gans et al., J. Chem. Phys. 150, 244304 (2019)], are reported in the present paper. Calculations at the internally contracted multireference configuration interaction level of theory have been performed to derive the rovibronic properties of the seven lowest electronic states of HC5N+. Equilibrium geometries, rotational constants, vibrational frequencies, electric dipole moments, and spin-orbit constants have been calculated and compared with experimental data when available. Adiabatic and vertical ionization energies from the neutral ground state as well as transition energies within the cation electronic manifold are predicted, using the convergence to the complete basis set limit. The accurate description of the complex energy landscape up to 32 000 cm-1 above the ionization potential allows us to perform Franck-Condon simulations of the photoionization spectrum to the X+ 2Π, A+ 2Π, B+ 2Σ+, and C+ 2Π states and allows us to simulate the A+ 2Π → X+ 2Π emission spectrum. The vibronic perturbations occurring on the excited potential energy surfaces are revealed and discussed, in particular, for the 3 2Π surface, which presents a double-well topography.