Q: Were King Cyrus and Darius believers or considered righteous in the eyes of God? I have read the accounts in the Bible, and it seems that they did believe in God, but does it mean they were saved?
A: Let me point out that in ancient polytheism, people were always happy to include and to adopt other gods; but adoption of other gods or new gods did not mean they rejected the gods they were already worshiping. They simply added Jehovah, the God of Israel, into their pantheon but did not make Him the only God. While Darius was willing to even admit that Jehovah was the great God or the greatest of the gods, he still did not reject polytheism but continued to worship multiple gods.
What people had to believe to be saved in the Old Testament is spelled out in Isaiah 43:10-12. First, they must accept that Jehovah, the God of Israel, was the only God — and if they believed that, it would rule out both polytheism and idolatry. Second, they had to believe the God of Israel was also the only Savior, which would rule out their own works for salvation as well. Both Cyrus and Darius failed on both points and in particular on the first point.