I'd lean towards Fourier analysis, but even as a disillusioned topologist, I can say their is some value in taking one topology course. In the grand scheme of things, one course isn't always going to matter that much. Topology can be useful in very subtle ways, even if it is not used explicitly. In your first topology course, you'll probably study point set topology, which I tend to think of as being almost the same subject as functional analysis (more relevant to physics), and historically, that was the way the subject developed. Topology builds your higher-dimensional intuition. If you want a deep understanding of physics, it's nice to be able to understand things like what a symplectic manifold is. The reason why Hamilton's equations aren't just meaningless symbols to me is that I know what a symplectic manifold is and Poisson brackets, the optical-mechanical analogy, and stuff like that. It's also nice to know things like covering spaces, and the fundamental group, since that sheds some light on the all-important double cover of SO(3) by SU(2). General relativity is another place where it would be helpful, more so if you go deeper into it (in my introductory GR class, the prof had to dance around the concept of a manifold because most of the class besides me didn't know what it was). It's not just in very specific research areas. Some basic topology is helpful in understanding the fundamentals of physics, just as long as you don't go overboard and become a topologist, like I did. Unless you want to be a string theorist or something, in which case, you'd be a bit crazy for wanting to be a string theorist. There are some other areas where knowing a bit more than the basics would be good, too--a physics student once showed me homotopy groups in some particle-theory book they were trying to read. But you could try to learn it on a more as-needed basis from "topology for physicists" books, like Nakhahara, later on, if you had to.
Fourier analysis is more fundamental, but I don't think topology would necessarily be a complete waste, depending on your goals and tastes. I'm not sure if you'd really need to know about Dirichlet kernels and Fejer kernels and delicate issues of convergence, much more than you would about topology, though. I think you probably come across what you need to know about Fourier series/transforms in the physics curriculum. As far as the more detailed stuff about Fourier series, I'd be a little wary of it if it isn't presented as well as Lanczos does in his book (Discourse on Fourier Series). It could easily become a tedious exercise in putting up with annoying technical garbage that would be mostly a waste of time. So, make sure to check out Lanczos from the library, just to make sure that doesn't happen.
