Dunno about you, but I think I'd be excitedly screaming with glee if I was riding an anti-gravity quad bike, even if my family passed away a couple of centuries ago when I was very long.
Characters can have tons of inner turmoil, but that doesn't mean they can't enjoy the moment. It's not out of character at all for her. I honestly don't get why people think her tragic back story should define her and render her unable to smile or laugh. She does both in Galaxy, and even has exclamation marks after some of her lines (such as when she first starts reading the storybook itself).
Also, I believe there was a part within the storybook itself that stated that Rosalina tries her best to keep up a happy face for the Lumas, even if she feels really sad.
Not that any of this matters in Mario Kart. Again, it would not be realistic if Rosalina DIDN'T excitedly scream and shout while driving at high speeds in quirky course.
Also, while you don't need to accept this as the truth - Miyamoto views the Mario characters as a cast of actors, to some degree. Not everything they do is legitimate, and the rivalries are just a show (or at least, the intensities of some are). I must stress that he only says he "views" the cast that way, not that we must do the same. If you really need a further point to justify Rosalina being happy and excited in Mario Kart, this makes perfect sense of it - in that she's an actress, and her role in Galaxy is her playing a part of a character. Outside of that, she could just be some hot model actress in the Mushroom Kingdom who everybody loves to see rock a motoring jumpsuit, and she's living the diva life.
The actual Miyamoto quote is:
I personally don't view Rosalina that way, I feel she is one of the completely legitimate characters who's back story is genuine. I just attribute her excited screams as well...her being excited. There's nothing saying she can't be that. And even when she is excited, she's more reserved than the rest of the characters.
Honestly, seeing Rosalina be happy and excited adds to her character to me. My problem with the main Mario characters in the main series is that they are often one dimensional. I don't mind them being happy, but when their only emotions are "happy" and "excited", then I tend to lose interest in the characterisation in the games. Rosalina displays a wide variety of emotions and has a deep enough personality for her happy moments to be more worthwhile to me. Plus, a happy Rosalina is a heart-meltingly adorable Rosalina, and there is nothing wrong with that. And it helps me discern the things she likes.
Also, Rosalina has a very special reason to be happy today: