No. The blow dryer is not a necessity. However, by omitting the blow dryer when styling you are severely limiting your styling options. The blow dryer is used to quickly set the hair and to add volume which elevates the mullet to the next level.
Opting for finger-styling only is a fine choice, but to get the most out of your mullet definitely look into purchasing a blow dryer and learning how to use it properly. It is a small investment that pays out immense dividends in the future if you’re a fan of grooming.