Weren't they in love in his past life?
This isn't much of a spoiler.
They were very close in the past life and i think one of them liked the other. Also, the princess had given up her own life in regret to protect the people. So it's pretty much that she's not a bad person and might be useful and also he considers her as a friend due to his past experience.
I think one of them liked the other but they weren't an official couple i believe. I could be wrong about them not being couple tho.
I’m so confused why does he want to save her so bad? It seems like a big hassle for nothing