There is no verifiable evidence to support the claim that the exposure to sun can cure blemishes. The opposite is more likely: exposure to sun increases exfoliation of dead skin cells which promotes clogged pores, the cause of blemishes. The only way to explain the root of this rumour is that a good suntan disguises the appearance of blemishes since it's harder to see them on slightly darkened skin.
Please be aware of the skin damage that comes with long exposure to sun without appropriate protection. Always use a sun protection factor of at least 20 when being exposed to it, ensuring sufficient protection against UVA and UVB rays. If you are anxious about applying oily substances on your skin you might want to choose an oil-free sunscreen or one for extra sensitive skin.