It's not rude but here's why i wouldn't do it (actually i wouldn't fax at all b/c they're all emailed now, but back in the day) :
this is actually when most people send their resumes (either when classes are out or when job's done). there's going to be a stack of them throughout the course of the night and a good chance yours will be at the bottom or caught in the middle. send it out first thing in the morning to give it a better chance of being on the top. HR usually sorts through that stack when they come in.