The declaration of Daniel 1.21 - “Daniel was there until the first year of King Cyrus” - is not an indication of Daniel's death.
Daniel 10.1 explicitly affirms that he still receives a revelation during the third year of Cyrus.
The first year of Cyrus must therefore be understood as a historical and literary landmark, not as the last year of the prophet's life.
This landmark has a particular significance.
Daniel 1 begins with Nebuchadnezzar, the intervention against Jerusalem and the transfer of part of the Temple vessels to Babylon.
It ends at the time of Cyrus, when the reverse movement becomes possible: the exiles can return to Jerusalem and the Temple vessels are handed over to be brought back to the city.
The chapter is thus framed by two great stages:
Nebuchadnezzar: Jerusalem → Babylon.
Cyrus: Babylon → Jerusalem.
Daniel lives through this whole period.
If the hypothesis of an age of about fourteen in 605 BC is accepted, he would be approximately 81 to 82 years old in the first year of Cyrus and about 83 to 84 in his third year. These figures nevertheless remain estimates and not explicit data of the book.
Daniel 1.21 finally has a major theological import. The first chapter begins with an empire that seems to triumph over Jerusalem and ends with Daniel still present when that empire has disappeared.
Kingdoms pass away; Daniel's faithfulness remains; and God's sovereignty runs through the whole of history.