No, the thief wasn't the first person saved; in fact he wasn't saved at all.
This erroneous idea that the thief was saved comes from the common mistranslation of verse 43:
Jesus answered him, “Truly I tell you, today you will be with me in paradise.”
The original Greek writing was written all in one case, without any punctuation.
That comma between "you" and "today" was inserted by the translators, based on their own understanding.
A few other translations place a comma differently, such as:
And Jesus said to him, "Truly, I tell you today, you shall be with Me in paradise."
— A Faithful Version
Jesus said to him, Truly I say to you today, you will be with me in Paradise.
— Lamsa Bible
This meaning better fits the Greek language structure, with "I tell you today" corresponding to the "when you come into your kingdom".
It also better fits the doctrinal teachings of the Bible.
Rather than leaving the thief in doubt about his fate, waiting for Jesus to return and perhaps remember him, Jesus assures him immediately that they will meet again in Paradise.
Many people blindly assume that "Paradise" is a synonym for "Heaven".
It isn't. Paradise is a large walled garden that will be placed on Earth during the Millennium, when Christ returns to Earth to rule his kingdom with the saints.
This Paradise in God's Kingdom on Earth is described in detail in the Book of Revelation.
“He who has an ear, let him hear what the Spirit says to the churches. To him who overcomes I will give to eat from the tree of life, which is in the midst of the Paradise of God.” ’
— Revelation 2:7
At the end of the Millennium, all the people that have ever lived and died without the opportunity for salvation will be physically resurrected ("the second resurrection").
They will live again in the Kingdom of God, here on Earth, and have their opportunity for salvation.
But the rest of the dead did not live again until the thousand years were finished. …
— Revelation 20:5
Remember, God wants everyone to be saved, even thieves and the billions of people that never had a chance to even hear the name Jesus, the only name by which one can be saved.
For this is good and acceptable in the sight of God our Savior,
who desires all men to be saved and to come to the knowledge of the truth.
— 1Timothy 2:3–4
The thief will soon experience death and will then immediately (from his point of view) be alive and well again in the Kingdom of God, where he will meet Christ, will learn and understand God's way of life, and will have the opportunity for salvation.
So no, the thief wasn't saved that day two thousand years ago.
But it is very likely that he will be saved in another thousand years from now.