My eyelid is swollen and hurts. should i ice it? or apply heat to it?

Depends on cause: Icing the lid help reduce pain and swelling. Once the swelling is improved, warm compresses help clear out the oil glands in the lids which often cause the redness and inflammation in the first place.

Probably stye: Most likely this is a stye which will respond to frequent applications of a very warm (not hot) washcloth. Try this for several days provided that the vision is not affected and there is no obvious pus. If persistent past 3-4 days then see your ophthalmologist.