Is the word "fair" a noun, adverb, verb, or adjective?
It's usually an adjective.
When you say "fair" (in the sense of "pale" or "just"), it is an adjective: fair maiden, fair decision, etc. The adverb form is "fairly," as in "I was treated fairly by the court."
Another definition of "fair" is "outdoor festival," as in "Spring Fair" or "Renaissance Fair." This is a less common usage, but it is a noun.
It can also be an adverb.
An adverb describes a verb, so in the sense of "Play fair" or "Be fair", fair acts as an adverb.