Lots to parse here.
First, the only translation I could find that contained "This day" is...
Douay-Rheims Bible
And Jesus said to him: Amen I say to thee, this day thou shalt be with me in paradise.
...the others say 'today'.
So, a minor point? Or not??
First; let's see what the thief said to Jesus:
39 And one of those robbers who were hanged blasphemed him, saying: If thou be Christ, save thyself and us. 40 But the other answering, rebuked him, saying: Neither dost thou fear God, seeing; thou art under the same condemnation? 41 And we indeed justly: for we receive the due reward of our deeds. But this man hath done no evil. 42 And he said to Jesus: Lord, remember me when thou shalt come into thy kingdom. 43 And Jesus said to him: Amen I say to thee: This day thou shalt be with me in paradise.
Luke 23:39-43 D-R
So the thief had a request; and he put a quailfier on it: "...when thou shalt come into thy kingdom."
So when Jesus replies, "this day" to him, just which day IS it??
Scripture tells us Jesus was 3 days in the tomb, and another place it says:
John 20:17 Jesus tells Mary ' Touch me not for I am not yet ascended to my Father; but go to my brethren an say unto them," I ascend unto my Father, and your Father, and to my God and to your God".'.
So just WHEN did Jesus come into His Kingdom?
Matthew 16:28 Amen I say to you, there are some of them that stand here, that shall not taste death, till they see the Son of man coming in his kingdom.
HMMMmmm...
Not too specific, but it narrows it down a bit.
Now this is where I'll stop; for many others, much more learned than I have 'discussed' this subject for years. The inquisitive can do the research on there own from here.
Good luck!