Learn Turkish: Dünkü anı çok özel. - Yesterday's memory is very special.

The suffix -ki attaches to time words to mean 'the one from that time'. Normally, -ki never changes its vowel to match the word before it. However, dün (yesterday) and gün (day) are the two major exceptions. Because they have rounded vowels, -ki changes to -kü to match them.

If the translation is 'yesterday's memory', why don't we use the possessive form like dünün anısı?

The suffix -ki turns a time word into an adjective. Because dünkü acts just like a regular adjective, it simply describes the noun without needing the possessive compound structure. It links to the noun exactly like saying güzel anı (beautiful memory).

Is the word anı just the word an (moment) with a possessive suffix added?

It looks exactly like it, but anı is a standalone dictionary word meaning 'memory'. You don't need to treat the -ı as a possessive or accusative suffix in this sentence; it is just part of the root word itself.

Where is the word for 'is' in this sentence?

Simple statements in Turkish often drop the 'to be' verb entirely for the third person (he/she/it/they). While you could add the suffix -dir to the end to make özeldir, it is completely normal to leave it off in everyday conversation. The 'is' is naturally implied by the sentence structure.