Everyone has their own style, and I admire you for wanting to maintain your style under difficult conditions!

Do what makes you happy, don't worry about what your friends say!

If you want to cut your hair for more reasons than just this one long bike ride go ahead and cut it. But if you are only thinking about this one ride, and you will be sad after the ride to have short hair, then don't cut it. I don't have long hair, but the suggestions for keeping your hair nice during your ride sounded good.