I am building up an old UO-8 right now. I was only going to put a front brake on it. Unless you are going to build this as a single speed, not a fixie, I think that is all you need. Do it!

BTW, what is the spacing on your rear dropouts? Mine is about 118mm clear, 120mm on center which means I have to spread the dropouts with just a little finger pressure to slide in a typical 120mm fixed gear wheel in place - but it seems to sit and track straight. I assume that my dropouts got mashed slightly at some point in the last 35 years and that is why they are a little off.