Is rarely a noun

No, the Word 'rarely' is not a noun. The Word 'rarely' is an adverb, a Word that modifies a verb, an adjective, or another adverb as not often; seldom.
Example: He is rarely late for lunch. (modifies the adjective 'late')

A noun is a Word for a person, a place or a thing.

Example: He rarely speaks of his father. (the adverb 'rarely' modifies the verb 'speaks'; the Word 'father' is an noun, a Word for a person)

The Word 'rarely' is the adverb form of the adjective 'rare'.

The noun forms of the adjective 'rare' are rareness and rarity.