How old was Belshazzar when he died?

Neither the Bible nor external texts give any date of birth for Belshazzar or indication of age so we can only speculate.

His grandmother, Adad-Guppi, was born 649 BC and lived 98 years (according the Haran Inscription). This being the case we expect his father, Nabonidus, to have been 60+ when he took the throne (555 BC). Belshazzar became regent of Babylon in 549 when his father left for Tema so he was presumably already an adult at this time. Putting this together, we might expect Belshazzar to have been 40+ at the fall of Babylon when he was slain (539 BC). But we have to acknowledge that this is speculation.