I would not say that it is crucial that you have a major in business or politics in order to learn on the program at all. But with that said, it might make it more interesting if you have a major in either of those subjects, something related, or have an interest in politics or European business practices or what is currently happening with the euro. I'm an international business major, and while I have many friends at home who I think would be able to understand pretty easily what was being taught, I doubt a lot of the ones with non-related majors really care about the history of the EU an the future of the euro. So I'd say if you are interested in the topic, go for it! Just know that most people will probably have a business, Economics or political science related major. This does not give them any advantage though other than possibly having more background in subjects that are discussed on visits and so they can ask better questions.
Like with any trip, it's really what you make of it! We had free time frequently, but even when we didn't in neat places that we might have only visited briefly, we found ways to make free time. You have to be willing to go non stop for a month. If you blink you might miss something! We would get up early sometimes before visits and go to museums and would stay up late in order to experience the culture in other ways. Some of my favorite visits on the trip were going to the British parliament in Westminster and getting to meet with a member of parliament and visiting the European commission in Brussels. It was a neat time to go to Europe with everything going on with Greece, the euro, and the EU. If you went this year (or anytime in the next few years I'm sure) it would be as interesting if not more because you'd be a part of so much. I felt like I was a part of history in the making!
Outside activities were too many to name. I can think of random things I really enjoyed, but also I remember the towns in general and really liking certain ones based on their individual cultures. Of course I liked Paris (but I'm a French minor), I also loved London even though it doesn't have the food culture that most other cities have, but what I really enjoyed were smaller towns like Strasbourg. It just had such a different feel but there was still a lot to see and do. What I would highly suggest is if there is something in a city that you want to do, don't let anyone stop you. If a group doesn't want to do something, go off by yourself and do it! Of course you have to do this within reason and be smart about this though. When we were in Paris, I really wanted to go to the Montmartre area and see the moulin rouge and the sacre coeur and nobody else did.. So I navigated the metro and went by myself. It was one of my favorite things, I think partly because it was a memory all of my own and it made me feel empowered.
But really, the people in your group are what make the trip or break it. The hard thing is you can't find out how everyone will interact ahead of time. I didn't know anyone before the trip. There were 17 of us and only 4 were girls! Our professor said normally there's a lot more girls than guys though. I became really close to a bunch of people on the trip and I even met up with 4 people in Chicago this fall for a little reunion. I don't think it would've been as good of a trip if the group hadn't gotten along so well!
So my overall advice: Go on the trip! Its a once in a lifetime kind of thing, and you get to visit places that you might never have access to go to otherwise (EU buildings, US embassy, etc.). Look up some background information before on the EU and the current happenings. And don't waste time; if there isn't anything arranged for a day, don't sleep in or just sit around. Go out and explore and experience everything!
Also, if I were to change anything about my experience, I would have stayed in Europe longer. While at first a month seems long enough, it went by so fast and there's still a lot of places we didn't get to see. Some people in the group went to other places in Europe before or after and I envy them! You already have a plane ticket over there, so might as well get a little more for your money!