I first met him ______ year ago.

Concept

English uses the indefinite article based on the SOUND that begins the following word, not its spelling. "a" precedes a word that starts with a consonant SOUND; "an" precedes a word that starts with a vowel SOUND. The definite article "the" marks a specific, already-identified noun, and no article is used when none of these conditions apply.

Application

Here the blank sits before "year ago", a non-specific time reference meaning "one year in the past". The noun "year" begins with the sound /j/ (a "y" glide), which is a CONSONANT sound, so the indefinite form is "a". "year ago" also introduces a brand-new, non-specific point in time, so an indefinite article is required rather than a definite one. The complete sentence reads: "I first met him a year ago."

Cross-check

Replace "a" with the candidate idea "one": "I first met him one year ago" carries the same meaning, confirming the indefinite article fits a single, non-specific year. The fixed time phrases "a year ago", "a week ago", "a month ago" all follow this same pattern.