I'm not going to keep my sister's job offer a secret
Who told you about your sister's job offer?
I found a letter and decided to visit her at work.
You shouldn't share other people's secrets. Your sister probably wanted to keep her job private. You shouldn't bring it up or visit her at work without permission.
Why not? I'll be a good brother.
You are not being a good brother by revealing a secret. You should ask your sister before telling anyone else just to be sure you are not going behind your sister's back.