It is impossible to know whether or not the “good thief” went “straight to heaven” or had to undergo Purgatory. We cannot really judge that. It could be that he died in a state of perfect contrition for his sins and received baptism of desire.
A larger issue is that the Greek manuscripts that contain this passage do not contain punctuation. So did Jesus say to him: “Truly, I say to you, today you will be with me in paradise” OR “Truly, I say to you today, you will be with me in paradise.” Punctuation makes a big difference.
Another issue is whether in this case “paradise” means heaven. It could also refer to the abode of the dead for the righteous, the bosom of Abraham. In which case the gates of heaven would not have been opened yet until the resurrection, so they would have been together in the abode of the righteous and the good thief still could have undergone some purgation before receiving the Beatific Vision. While this is a highly contested theory of the use of “paradise” it is still debated.
And, most importantly, since the cleansing of Purgatory takes place outside of space and time as we currently know them then the concept of today, tomorrow, or whenever in relation to Purgatory is not a one-to-one analogy.
Generally speaking the most common understanding of the passage has been that the good thief went “straight to heaven” due to his perfect contrition for his sins, but some other theories are also acceptable to the faith.