The 501st spent most of there downtime on Coruscant for various reasons, all of which involved Anakin Skywalker in some form or another. Most of the troops didn't mind. They would go to 79s and catch up the whichever of their brothers were currently stationed there.
Rex, however never really enjoyed the downtime. He, like many of his brothers, hated being inactive, he always had to have something to do. Because of that, downtime was the last thing he wanted.
That changed when he was given his violin. Suddenly Rex started looking forward to the hours of free time he had on Coruscant, where he had nothing to do but play his violin.
When they arrived and his work was finished he would change into the civilian clothes he managed to procure, put on a low brim hat, grab his violin and went off to find a corner of Coruscant to play.
He never played at the same place twice, moving up and down the levels, not caring about the shadier characters moving around some of them. (Months after he first started playing, he found out he had a following that made it into a game of trying to find out where and when he would appear.)
Rex loved his time on the streets of Coruscant. During that time he was not a soldier fighting for whoever held his leash, he wasn't a Captain responsible for the lives of hundreds of men, he wasn't a clone to be hated or thrown away. During that time he was just another street musician playing his violin for whoever cared to listen.
So when Cad Bane himself once stopped to listen and throw a few credits into his case, Rex didn't give chase or thy to arrest him. No, Rex just let him walk away, just like he did any other criminal that came his way, without a care for what they've done or what they'll do.
Surprisingly he learned a lot on the streets. He learned to pick locks from one particularly appreciative listener, from another he learned slight of hand, from another how to hide in plain sight, and another gave him a data pad of different languages he should learn.
Rex also found a lot of information. It's almost surprising what people are willing to tell a musician. He learned about who had bounties on their heads and why, who worked for who, who was important and why, who to never approach, who would do what for the right price, and so much more.
His time on the streets were valuable in more ways than one. He was so surprised the first time someone threw credits into his case that he almost stopped playing. None of the clones got paid, so for a stranger to so casually give him money for no apparent reason was startling at best. And people kept doing it.
At the end of his first day playing the streets of Coruscant he had enough money to buy a meal and two pairs of clothes (so he didn't have to borrow any anymore), and he still has some leftover.
He never gave any of the money away (he told himself that it was just in case something happened, not because he had a bad feeling that he would need every credit he could scrounge together). Instead he hollowed out space in his violin case and hid the credits in there, exchanging them out for the next biggest denomination when there was too much to fit.
It wasn't just credits that he was given. He received various odds and ends from machine parts to medicine to herbs to spice and so much more. He kept what he could fit into his case and sold the rest. Eventually he got a bigger case for his violin with various secret compartments to stash his earnings.
Funnily enough, despite being a street musician, he still got dragged along on crazy adventures, to the point where he started thinking that maybe it wasn't always the Jedi's fault that things went sideways (he threw that thought away the next time he saw his General and Commander doing something stupid, it was completely their fault).
Some of the adventures ended with him helping a robbery, stopping a murder, getting caught in the middle of a lovers spat that ended when all three of them in bed together, collected a bounty, and getting a tattoo on his back- it said 'Part Attack Dog, Part King' with a crown wrapped in chains- all on the same day.
Rex was rather thankful that none of his men asked any questions when he would show up with some blood on his shirt, lipstick on his neck/face, or smelling like smoke and alcohol. They probably assumed he was doing what most did, went to a bar to drink or looked for a bed partner, not that they were completely wrong. Still they wouldn't believe half of what he got up to despite working with Jedi.
Overall Rex would have to say he really enjoyed his time of to do 'nothing of importance'.