No. Cellulitis is an infection of the skin and subcutaneous tissues. Neosporin does not penetrate below the surface and will not help cellulitis. Oral or intravenous antibiotics are usually needed.
What's the cause ? Cellulitis is basically a visible inflammation (itis) of the skin and soft tissues (red, hot, swollen and tender) . It can be caused by many things including sunburn (radiation), trauma (sprained ankle) or poor vein circulation (stasis dermatitis). It need not be caused by bacteria. If you have repeated episodes of this, get to a wound care or vascular specialist, the diagnoses you have is wrong.