Is the pain in my sty supposed to spread to my eyelid.?

Sometimes: A stye is like a pimple on or in the lid. A pimple on the face, because of deep skin attachments stays in one spot. On the lid, there are no deep attachments because the lid requires mobility, so the inflammation can diffuse over a broad area of the lid. So it commonly will. Treatment is best started with topical very warm packs applied frequently. If that fails then see your ophthalmologist.

Stye: If the stye is worsening despite warm compresses you should be evaluated by an eyemd. Infection can sometimes spread and require treatment with antibiotics. Some also require drainage in the office.