Before the Spirit of the LORD rushed upon Saul, he worked for his father. Afterward, he gradually took on the responsibility of being the first king of Israel. He changed from being a plain citizen to being the king.
The change started immediately. 1 Samuel 10:
9As Saul turned to leave Samuel, God changed Saul’s heart, and all these signs were fulfilled that day. 10When he and his servant arrived at Gibeah, a procession of prophets met him; the Spirit of God came powerfully upon him, and he joined in their prophesying. 11When all those who had formerly known him saw him prophesying with the prophets, they asked each other, “What is this that has happened to the son of Kish? Is Saul also among the prophets?”
The change was observable as signs. Saul prophesied.
The next observable behavior was when he decided to rescue the city of Jabesh.
1 Samuel 11:
5Now, behold, Saul was coming from the field behind the oxen. And Saul said, “What is wrong with the people, that they are weeping?” So they told him the news of the men of Jabesh. 6And the Spirit of God rushed upon Saul when he heard these words, and his anger was greatly kindled.
Saul showed the righteous anger of God. He led the Israelites to defeat the Ammonites.
11 And the next day Saul put the people in three companies. And they came into the midst of the camp in the morning watch and struck down the Ammonites until the heat of the day. And those who survived were scattered, so that no two of them were left together.
After this success, Saul had proved himself and his kingship was undisputed.
15So all the people went to Gilgal, and there they made Saul king before the LORD in Gilgal. There they sacrificed peace offerings before the LORD, and there Saul and all the men of Israel rejoiced greatly.
What does it mean that the Spirit of the LORD rushes upon someone and turns them into a different / another person?
It means that the Spirit turned Saul from a man of the field into a person suitable to take the kingship.
What exactly is changed?
His heart in the Hebrew sense of the word which includes the feelings, the will, and the intellect.
What was different in Saul?
Before, he was leading donkeys. After, he was leading men.
When this happened to him?
The change started soon after Samuel anointed him.