Is periorbital cellulitis curable?

Yes. Periorbital cellulitis is curable if the antibiotics are effective against the organism (bacteria usually). This can be serious and needs to be managed by a physician possibly in the hospital.
Emergency. This is a very serious problem and requires an immediate trip to the er. This is an infection of the tissues surrounding the eye and can spread to sinuses or even to the brain. You could lose vision, nasal function or even your life if it gets into the brain cavity. Go for help without delay.