I actually like your translation, but to say that it is semantically and grammatically accurate is open to debate. Conceptually "quest'anno" here answers the question when, not what/which (year) as in your sentence. "Finisce" is intransitive, its subject being "il secolo", in technical parlance this is referred to as an ergative (or unaccusative) verb (btw, that is why its subject normally follows the verb in Italian, cf., "Arriva Gianni" = John arrives). Yet, in your translation, the verb "concludes" is transitive, its subject being "this year" (as a numerical entity). The difference between transitive and ergative verbs is like the difference between "I broke the window" vs. "the window broke" in English. Your translation expresses an equivalent idea and is valid in that sense, but if you want to use translation as a tool to acquire a language, it is better to think of this Italian sentence as "this year, the century ends."
Shouldn't "Questo anno" be also accepted? I was told that "questo anno" and "quest'anno" are the same, but due to the fact that "questo" ends with a vocal and "anno" starts with one, it's difficult to pronounce, thus it is abbreviated,but grammatically both should be correct. Is this wrong? Thanks!
Part of the grammar rule is that since "anno" starts with an a, it has to be "quest'anno," so grammatically "questo anno" would be incorrect, making that answer wrong in duolingo. Just like if someone asked you to hand them "a apple." That's grammatically incorrect since we have a grammatical rule to change "a" to "an" when dealing with vowels. Does that make sense?