This ranks fairly high on my list

Is rank a copulative/linking verb in This ranks fairly high on my list ?

What Part of Speech is high here?

High (adjective) https://www.oed.com/oed2/00106032

High (adverb) https://www.oed.com/oed2/00106033

Rank (verb) https://www.oed.com/oed2/00197225

In the Oxford English Dictionary (OED), we find the following in the entry for the adverb high :

In a high position; at a high altitude; far up, aloft.
Often overlapping with or difficult to distinguish from the adjective used as the complement of a verb.

The sentence

[1] This ranks fairly high on my list.

would seem to be one of those cases where one could fairly interpret high as either an adjective or an adverb.

If, in your sentence, high is interpreted as an adjective, then indeed you could say that rank functions as a 'linking/copulative' verb—if by that you mean that it takes a PC as complement (as BillJ said in the comments).

On the other hand, if, in your sentence, high is interpreted as an adverb, then rank would not be a 'linking/copulative' verb—if by that you mean that its complement is not a PC.

Discussion

With some other verbs, there is no ambiguity. For example, high is definitely an adverb in

[2] He climbed high on the list of the state's most sought-after criminals. (source)

because climb cannot take an adjective phrase as a complement. For example, it's he climbed beautifully, not ^*he climbed beautiful.

And it is definitely an adjective in

[3] Courtesy never seemed high on the list. (source)

since seem takes a predictive complement (PC) as a complement, and a PC must be a noun phrase (NP) or and adjective phrase (AdjP). And so it's he seemed beautiful, not ^*he seemed beautifully.

But rank can take either an adjective phrase (AdjP) or an adverb phrase (AdvP) complement:

[4]  i  There are many ways not to be a sportsman, and
          neglecting limb lines and trot lines ranks highly      [AdvP as complement; (source)]
          on the list,

       ii  We in the Air Force consider that it ranks equal
           in priority.                                                                        [AdjP as complement; (source)]

And since high can be either an adjective or an adverb, it is ambiguous what kind of complement it is in [1].