Is there an old, rarely used word which means "an archaic word"?

Ancient is used quite often, but at least talks about it being old.

There is also the term obsolete meaning no longer used. Oxford's COD has obsolete as disused, discarded, antiquated.

Walter W. Skeat uses the term obsolete in his work on etymology, A Concise Etymological Dictionary of the English Language.

an obsolete word

If you’re partial to Greek, then depending on what you’re aiming for, you might consider any of:

• palaeonym (old word)
• archaeonym (ancient word)
• cryptonym (hidden word)

Or the more cryptic:

• spanonym (rare word)
• lanthonym or xechasmonym (forgotten word)
• chamaeonym (a lost word, one that’s on the ground)

If you want most people to be able to guess what you mean, use one from the first set; if you don’t, then use one from the second.

If those won’t do, you’ll have to seek out some of those words’ more recherché poecilonyms.

Maybe ‘aureate’?

Is it possible you were thinking of ‘aureate’? It starts with ‘A’, it's an adjective, it's decently old, it seems decently autological, I don't think it's a common word, and here are some quotations from Wiktionary's entry on it (copied under CC BY-SA 3.0 terms; bold emphasis mine):

1996, Keith D. White, John Keats and the Loss of Romantic Innocence [Costerus, new series; vol. 107], Amsterdam; Atlanta, Ga.: Rodopi, →ISBN, page 94:

In the only monograph on the subject, John Cooper Mendenhall describes aureate terms as "words designed to achieve sententiousness and sonorous ornamentation of style principally through their being new, rare, or uncommon, and approved by the critical opinion of their time." Since the time of Lydgate, who named these loan words and neologisms "aureate terms" to denote their linguistic gilding, Latin or Latinate words were considered the prime examples. However, readers who found aureate terms pretentious began to call them inkhorn and inkpot terms, both references to the receptacles scholars carried to hold ink.

2002, Simon Horobin; Jeremy J. Smith, An Introduction to Middle English, Oxford: Oxford University Press, →ISBN, page 73:

Aureate vocabulary is derived largely from Latin, although some words have a French basis; it was devised as a 'high' or 'elevated' poetic diction used for special ceremonial or religious occasions. Perhaps the best-known practitioner of aureate diction in the late ME period was the poet John Lydgate (c. 1370–1449/1450), monk of Bury St Edmunds, court poet and self-styled disciple of Chaucer. Something of the flavour of Lydgate's aureate verse may be captured in the following extract from his A Balade in Commendation of Our Lady (a poem, incidentally, where Lydgate calls for aid from the auriat lycour of the muse Clio – Lydgate seems to have been the first English writer to use the term 'aureate').

_{—“aureate”, Wiktionary, The Free Dictionary, 20 June 2020, https://en.wiktionary.org/w/index.php?title=aureate&oldid=59570204 (accessed 21 Oct. 2020)}

Latin is certainly old, even archaic, and is largely now unused, and thus I would say that ‘aureate’ seems decently autological in that it itself is Latinate:

Origin

Late Middle English from late Latin aureatus, from Latin aureus ‘golden’, from aurum ‘gold’.

_{— “Aureate | Definition of Aureate by Oxford Dictionary on Lexico.com also meaning of Aureate”, Lexico.com, Oxford University Press, 2020, https://www.lexico.com/definition/aureate (accessed 21 Oct. 2020)}

(Despite the ungraceful wording of that page title, that really is the Oxford University Press that publishes the Oxford English Dictionary.)

Wikipedia, albeit with some citations needed, gives an example of old, aureate Scottish poetry that itself uses the word (also CC BY-SA 3.0; emphasis all theirs):

An example of considered diction with an aureate inflection occurs in the Scots couplet

Up sprang the goldyn candill matutyne,
With clere depurit bemes cristallyne

— William Dunbar, The Goldyn Targe, lines 4–5

Matutyne, depurit and cristallyne are aureate words. Aureate diction occurs in the noun phrase golden candle matutine, a circumlocution which stands for sun. The couplet can thus be translated as: up rose the sun with clear pure crystal light.

Dunbar himself uses the term later in the same poem in a passage that employs the limits to expression topos. It occurs as part of a dream vision in which the makar is describing the army of goddesses he has witnessed alighting upon the earth:

Discrive I wald, but quho coud wele endyte
Hou all the feldís wyth thai lilies quhite
Depaynt war bricht, quhilk to the heven did glete?
Noucht thou, Omer, als fair as thou could wryte,
For all thine ornate stilís so perfyte;
Nor yit thou Tullius, quhois lippís suete
Off rhetorike did in to termés flete:
Your aureate tongís both bene all to lyte
For to compile that paradise complete.

— The Goldyn Targe, lines 64–72

I would (attempt to) describe (the scene), but who could satisfactorily frame in verse the way in which all the fields were radiantly adorned by those white lilies (the landing army) that shone upwards into the sky? Not you, Homer, sublime as you were in writing, for all your faultlessly ornate diction; nor you, Cicero, whose sweet lips were so consistently lucid in rhetoric: your aureate tongues both (the Greek and the Roman) were not adequate to describe that vision in full.

_{—“Aureation”, Wikipedia, The Free Encyclopedia, 12 April 2018, https://en.wikipedia.org/w/index.php?title=Aureation&oldid=836085342 (accessed 21 Oct. 2020)}