Well, first and foremost, I find it very unlikely that Katniss wouldn’t volunteer. Everything we know about her character indicates that she would unless there was some kind of restriction—she was too old, no volunteers were allowed, etc. So unless we’re changing the nature of Katniss’s character (which would in turn change Prim’s character, to some degree, considering that Katniss took care of her for the last several years of her life), Katniss would volunteer. You can’t convince me otherwise.
But, okay, let’s say Katniss is too old or there is a rule change for something like a Quell or we’re in a universe where no volunteers were ever allowed. Prim goes into the Games. We know that Prim is really clever, but that she was also pretty hopeless in the woods. She didn’t even want to hurt bunnies. I think she would be pretty good at finding edible plants, considering she has background with this, but that’s not going to get her that far. Unlike Katniss, she doesn’t know where to go to find water, how to climb trees (that we know of), how to even cover her tracks.
Prim’s entire character is built on being a healer: she is compassionate and determined to protect and heal and give life, not take it. I think her sense of self-preservation would help her survive for a bit, and she definitely wouldn’t be stupid enough to chance the Cornucopia so she’d probably make it at least a couple of days. But placed in a combat, I don’t think she’d have the fighting skills or instincts to make it through. I think Prim simply wouldn’t have the will to murder anyone. Katniss knows this about her, it’s one of the reasons she loves her so much and would do anything to protect her.
Also, not least of all, Prim is tiny. Katniss’s hand-me-downs don’t even fit her. And although she’s more well-fed than most of the Seam kids her age she hasn’t had the rigorous exersize her sister has to develop her strength and stamina.
And sure, she’s cute and that might garner sponsors (although without Katniss volunteering there is nothing to make her special except maybe a flamey costume by Cinna, and hey, Rue was cute and that didn’t really work out in her favor), but sponsors can only get her so far.
TLDR: Prim would die, and Peeta would almost certainly never even have to worry about killing her.
You know I wouldn’t be able to say no to this one. Someday, Kika. Someday.