first-rate vs very well

first-rate is an adjective but very well is not an adjective.

Word	Noun	Adjective	Verb	Adverb
first-rate	No	Yes	No	Yes
very well	No	No	No	Yes

As adverbs, first-rate and very well are synonyms defined as:

first-rate (adjective)	very well (adjective)
of the highest quality

first-rate (adverb)	very well (adverb)
quite well	quite well
	an expression of agreement normally occurring at the beginning of a sentence