As a new member in the family of two-dimensional (2D) transition-metal dichalcogenides, the excited-state properties of Janus MoSSe have not been correctly understood yet, especially in the consideration of temperature effects. On the basis of many-body Green's function perturbation theory, strong excitonic effects prove to dominate the absorption, with a large binding energy of 0.85 eV assigned to the lowest A exciton. As a consequence of atom zero-point vibrations, electron-phonon coupling decreases the band gap of Janus MoSSe by 40 meV even at 0 K. As the temperature increases, the excitonic peaks (position, intensity, and broadening) change distinctly. At room temperature, the lowest A exciton is positioned at 1.59 eV, in acceptable agreement with the photoluminescence observation of 1.68 eV. For Janus MoSSe, real-time simulations indicate that the buildup of the intralayer exciton can be finished in an ultrafast time scale of 212 fs. Our results are fundamental for correctly understanding the excited-state properties and light-matter interactions of 2D materials, particularly for Janus MoSSe-based van der Waals structures that hold great promise in applications in optoelectronics, photovoltaics, and valleytronics.