While many Greeks and Romans worshipped a large pantheon of gods, specific temples with its own cult dedicated to a chosen deity, were a well-known phenomenon. These cults arose as a sort of reverence for certain attributes of each deity. The Greeks’ admired physical strength as well as Atlas having connections to astronomy and celestial bodies. The Romans had a penchant for creating many unique cults, even worshiping real figures as deities (such as Antinous). Despite being a figure associated with the Titans and thus punished for it by the gods, are there any known examples of a Cult of Atlas, or worship of any other “punished” figures of Greek and Roman mythology?
No, no cult sites devoted to Atlas. When he appears in pictorial arts, his role is ancillary to the story of Herakles. Also, there are no locations that really lend themselves to a cult centre: Atlas is put in different locations in different sources. Sometimes in the Atlas mountains (hence the name), but sometimes in Asia among the (fictional) Hyperboreans.
There are a few complications which tend to get obscured by the category 'Titan':
(1) He belongs in a pre-Olympian generation, so in some ways it makes sense to group him with the Titans, but it isn't clear-cut the way it is in Percy Jackson. In many contexts the 'Titans' were often just three specific individuals: Kronos, Iapetos, and Okeanos. Sometimes, other divinities of the same generation get lumped in with them, and get called Titans. Sometimes.
(2) Some other divinities of the 'Titans' generation are still hopping around without being punished in classical myth. In Homer, Hyperion is still in charge of the sun, and Dione is hanging around on Olympos. Mnemosyne is regularly invoked by poets, Hekate seems to count as an 'ex-Titan' in some contexts, and Prometheus is still kicking around for a while after the war with the Titans.
(3) Even one of the three canonical Titans, Kronos, had cult-sites. The Kronia festival in Athens was a fairly major event, for example.
(4) Atlas' job isn't a punishment. It's simply what he does. It isn't a pleasant task, it is forced upon him, and he does ask Herakles to relieve him. But there's no crime that he's being punished for. Hesiod does talk about it as a task that is forced upon him (Theogony 517); and in the play Prometheus bound, 347-352, Prometheus expresses sorrow for his brother --
For even if I am unfortunate, that wouldn't be reason for me
to want everyone to meet horrible fates.
Absolutely not. In fact, the fate of my brother
Atlas pains me, who in western regions
stands holding the pillar of the sky and earth
on his shoulders, a pain in the arms.
But there's nothing more specific than this. He's just this guy that has a very unpleasant job imposed upon him. There is one exception: a Roman-era mythographical text, Fabulae (a.k.a. pseudo-Hyginus) 150, casts Atlas' job as a punishment for being the ringleader of the Titans. But that's clearly in contradiction to what other sources say about the Titanomachy; it must be a retcon.