I woke up this morning with my eyelid hurting, specifically while applying eyeliner. My eyelid is sore to touch & it hurts when I blink. Is it a stye?

Probably. Apply hot compresses and see what happens over next day or 2. Also could be a bite, or early chalazion. If you wear contact lenses, see doctor today, though.
Likely. Stop using the makeup and apply a warm wet washclothe at least four times per day. If this does not work in a week., see an ophthalmologist.