AHC: Well remembered Caligula

As the title suggests, find a way to have Caligula be remembered as a good emperor, not a great emperor necessarily, but an emperor who's deed's that were beneficial to Rome are more well remembered than any actions which were harmful to it, preferably without resorting to the good old fashioned "he dies before going crazy."
My lecturer on the Principate supported the theory that Caligula only went ‘mad’ after he discovered the letters and correspondences related to the various atrocities that were committed on his family by Sejanus and his cohorts, completely losing faith in the Senate and thereafter beginning to exact his revenge on them. A youth spent in the close and prison-like vicinity of the man that executed his brother Nero Julius Capsara before his eyes probably doesn’t help either. I’m looking at you Tiberius.

The best (but cheaty) way to avoid his ’insanity’ would probably be to make sure those correspondences somehow disappear and a lot of the people who were revealed to be part of the conspiracy get away with their crimes. That way the 6-month honeymoon period of the ‘perfect’ princeps never ends and Caligula might go on to be the youthful prince that was promised to the Roman people.
That way the 6-month honeymoon period of the ‘perfect’ princeps never ends
The problem with that though was he was spending an unsustainable amount of money, even before the illness. He was able to alleviate that somewhat by taking money from his political enemies, but even then it still wouldn't be enough.
I mean, if he had died when he became seriously ill during his early reign, he'd be well remembered by the people of Rome compared to OTL, though he'd probably be quickly forgotten by most people today, as he'd only have ruled for a few months. He'd basically essentially be the William Henry Harrison of Roman Emperors.