How can you get pink eye if you don't have a cold or sinus infection?

From someone else. If you yourself are not infected with a cold or upper respiratory infection, then you most likely came into contact with someone or something that was carrying the virus. Once you came into contact with the virus you touched your eye and infected it.
Virus only in eye. If a virus is introduced into the eye, one can have a viral infection of the eye without a cold.
Pink eye. Not all pink eye is viral...It may look like pinkeye but be caused by allergy or a chemical or dry eye or herpes virus. Classic pinkeye in adults is usually caused by a cold virus, though...