I'm pretty sure the writing on the stick is a generic advertising gimmick, but I've never actually seen a Japanese popsicle so I wouldn't know if it's the common thing or not.
I do know it's pronunced "ataru" and can mean "to hit/to strike", "to shave" and "to be called upon by the teacher", among other things.
I'm also not sure if she's saying two, sun or a Monty Python reference.
It says, "Winner," because in Japan, most popsicle companies have a thing where they print, "Winner" or "Lose" on the stick. It basically is redeemable for another popsicle.