I am no bowser main but I have really taken a liking to him in this smash. I really enjoy playing with his new move-set and stuff, therefore now and then I go into "For Glory" with him. Today I fought a very good Rosalina and was having lots of trouble with my mains, however as I switched over to Bowser, I quickly two-stocked her on the first match. After that the matches were rather even. So I wouldn´t say Rosalina is bad news to Bowser.
I would suggest rushing and grabbing, rushing and grabbin all day long. If they have Luma out, might as well pivot-grab instead as they can break free out of grabs because of the pesky star. Also, gimping... I had the easiest time killing Rosalina when I went for the gimp, otherwise she WILL return and once she grabs the edge it is bad news as she will once again be grounded.
On another note... dem dash-attack. Her dash is pretty annoying, having two hitboxes (more with Luma) so you need to time your shield-grab, otherwise you will get hit by one of the later hit-boxes, however, once you get the hang of it it becomes pretty easy and rewarding to punish her dashes.
Mmm, what else?... Grounded up-B is good for racking up damage and maybe killing, rely on F-tilt and jabs to rack up damage and knockbacks... also fair is pretty good once you have her in the air. Good thing is that Bowser is pretty stubborn and will not die anywhere quickly, so you can wear them out with a little patience and momentum. Rack up the damage with grabs, fire, jabs and ftilts and then deliver the KO with fair or up-B. This would be my suggestions.