I feel sympathy for the 1st prince. He is the true victim in all this. He is actually a nice kid. He has been told his father is the emperor and wanted to have that parental love, as his mother barely gave him any. And he just wanted some love, but he could feel some distance from the emperor. So he was jealous when Caryle came into the picture because the emperor gave him love. The 1st prince didn't know why he was given coldness but this kid that popped out of nowhere received it all.
However, he did try to be a brother because Caryle was a nice kid and was calling him brother. The 1st prince started to feel he had family. He started trying to get close and gave Caryle sweets, not knowing it was poisoned by his mother. He felt so guilty about it. Even though it wasn't his fault. After his mother death, he found out he wasn't the emperor actual son. He understood why he never gotten the fatherly love he wanted. He agreed/accepted to fake his death, because it was known the 1st prince poisoned the 2nd prince, and promised to never come back.

please spoil me what happened to the first prince he isn't the child of ml then what happens to him after