In winter it is definetely cold in the mornings, much more so after rinsing with cold water. Most days I need to cover my head although I hate to do, specially when freshly shaved.
But it is advisable to always have a cap with you, if you stay outside a lot even the sun in winter can harm your skin, not to tell what can happen in summer.
I apply tons of sunscreen, always carry little samples with me, at least SPF30 if not 50