Rosalina is the hardest character in the game to grab. Luma's entire job description is to be in your way. Getting rid of him is your top priority whenever he is alive. Rosalina's punish game on her own is reasonably weak, and even more importantly, she can't seem to kill us on her own. Rosalina's shield and dodges don't protect him. Dash attack, falling on them with Nair, and fortress get some excellent mileage here for separating the two, just don't use them all the time. Aiming Ftilt or Dtilt generally works out since we don't clank with his jabs or rapid jab. You can spray Fire Breath as well, but she doesn't really have issues getting over it. FB is interesting because it stops luma in his tracks as you deal damage, so I would only use it when he's approaching me with jabs. And if luma has sustained no damage, he doesn't get launched from Dtilt's second hit, letting you string the first hit of another. In a real match with a fresh Dtilt, the first two are enough to deal 40% damage, which is most of his health in one second before launching him. And if you do somehow manage to grab rosalina and you're near a ledge or atop a platform, Uthrow can kill a nearby luma instantly by semi-spiking him to the right or left. It won't happen often, it's just really fun to watch. If you grab her with Flying Slam you can also land on Luma to damage him (15%) and then continue your attacks on him before he regroups with Rosa.
Once she's alone, she'll want to run from you. If she takes to the air, you might want to consider staying grounded and covering her landing, or the ledge if that's where she's retreating to. Our aerials aren't terrible in terms or range and damage, but hers can be much better depending on which direction you're approaching her. Especially if we're directly above or under. Respect those moves.