Are corneal ulcers are painful?

Generally so: These are focal deposits of growing bacteria on the surface of the cornea. The eye reacts strongly, gets very red and painful and usually is quite light sensitive. The vision almost always drops. Most come from contact lens wear problems. Remove your lens and see your ophthalmologist right away.

Nerve irritation: The cornea is richly innervated with nerves of pain. An ulcer stimulates those nerves and a strong signal for pain is produced. Treatment is generally highly effective so 'honor' this pain by seeking help immediately.