Apparently, Yester cannot be used alone in a sentence, except when accompanied by "day (yesterday) or year (yesteryear)". It cannot be used incombination with other portions of time like; yestermonth, yesterweek, yestersecond, yesterminute, yesterhour or yestermillisecond. So what is "Yester"?
"yesterday" is related to German gestern (the last day/ day before today) and Latin heri. From heri an adjective form was derived: hesternus. Change h in hestern.. to g and you get gestern or change h in hester... to y and you get yester. (This play with letters is a simplification. English y surely did not develop from Latin h.)
The etymology of heri is not clear. There are a lot of old Indo-European variants, but they are mere variants and there is no idea of what might be behind heri. My idea would be that Latin heri might be related with the Latin verb form fuere, then the idea behind heri would simply be: the day that was.
So the question "What is a yester? may be asked, of course, but as you see some words have a development of two thousand years and more and not every word element was a noun.
Further to rogermue's answer: according to Online Etymology Dictionary, yesteryear (n.) was
coined 1870 by Dante Gabriel Rossetti from yester- + year to translate French antan (from Vulgar Latin *anteannum "the year before") in a refrain by François Villon: Mais ou sont les neiges d'antan? which Rossetti rendered "But where are the snows of yesteryear?"
It was a nonce word, obviously modelled on 'yesterday'. It caught on. Many don't.
What might be called the literal sense (last year) is, however, rarely used:
yesteryear (plural yesteryears). (poetic) Past years; time gone by; yore. (rare) Last year. [Wiktionary]
"Yester" is what's called a "cranberry morpheme", (like the "cran" of "cranberry") meaning it has no meaning on its own but serves to distinguish words.