You could wear a baseball cap right way round under your helmet to keep the sun off your nose and keep the perspiration out of your eyes, and sew on a piece of hanging cloth, French Foreign Legion style, at the back to keep the sun off your neck.
l wear a cotton dress shirt with the collar turned up to meet the helmet, and the cuffs overlap the leather dress gloves I cycle in. Works for me.
BTW, I hold my cotton khakis up with braces so that air can flow through, rather than a belt which would restrict airflow, and wear the shirt outside the trousers to promote airflow.
You'd look a right neddy in that hat George's chum uses, especially in red. A free air ticket to Australia should come with that hat.
I'm so fair that I never tan, my skin only burns. Recently I've had surgery for two melanomas and picked up a valuable tip. One of the surgeons told me that if I wanted all traces of the operation on the back of my hand to disappear, I should rub on Vaseline daily. The Vaseline has also had additional benefits beyond disappearing all traces of the surgery: it has almost eliminated the dried skin spots which are the starting point for melanomas when they harden (and faster if you scratch them off--once they start it is a lose-lose lottery), and the red blotches and brown spots of age have gone, so that my hands look forty years younger. I'm not vain but...