Hi L.,
I have four children. I think each birth or addition changes your child, from personality to their identity. This is not a bad thing, it's just true. Each time I had a baby, I would tell the big brother, "Oh (baby) loves you! She is going to be your buddy when you grow up! She thinks you are so wonderful!" And I spend a great deal of time one on one telling my children how wonderful having a sister or brother is. I tell them, "When you grow up and you are really sad, you are going to call your sister. She is going to be there for you! Be nice to her!" I tell them what it was like to be a big sister, little sister have a big brother little brother when I was little etc. And I tell my oldest when she loses her patience, "Whether you like it or not, you are the big sister. You need to treat your brothers the way you want them to treat you. They learn from you. If they hit you, tell them you would never hit them. And I tell her, "You don't want your little brothers to grow up and remember you as a mean sister do you? Don't you want them to remember how you helped them and cared about them and played with them?"
Last, it is really tough! Nothing surprises me anymore. My son hit my daughter with a heavy Thomas the train bridge once!" Now that I have had four children, I realize, all children will hit, bite or do what ever it is you think they would never ever do. It doesn't mean they are bad. It is our job to explain why that is unacceptable and mean. It takes telling them over and over and modeling and slowly they learn empathy and their conscience kicks in. I try not to ask why: Why did you hit her?!!! Instead I say, "There is never a good reason to hit anyone. You need to use your words... etc..."
Kids learn to cope with their feelings and act appropriately, and they need your guidance as they build that skill. I would try to limit his time with the baby completely alone. And praise him for his good big brother behavior as much as possible. Remind him of all the great things he can do, because he is big. He can run and go on the swings, etc. And he will get to teach his sister all those things.