At the end of RotJ, Han was about to concede to Luke over Leia and was giving her his blessings when Leia revealed that Luke was actually just her brother. What did Han think and feel upon that revelation?
He’s surprised and pleased
In canon, he appears to have experienced surprise, followed by distinct happiness.
“Oh. No, it’s not like that at all. He’s my brother.”
“Oh,” says Han, then pauses to think what this means. And then pauses a little longer to think what it all means. “Oh!”
And they kiss. And it is such a meaningful kiss, such a passionate kiss, that I’m relieved to say that it is soon interrupted by Wicket hurling himself at them with an ecstatic “Meeeercheeewawaaaa!!!!”
Beware the Power of the Dark Side!
Moving down the canonicity ladder to Legends novelizations, he is said to have been "stunned, embarrassed, and elated."
Then she realized what he was talking about. ‘Oh, no. No,’ she laughed, ‘it’s not like that at all - Luke is my brother.’
Han was successively stunned, embarrassed, and elated. This made everything fine, just fine.
Return of the Jedi