Learn Russian: Он был так занят, что не позвонил мне за день до встречи. - He was so busy that he did not call me a day before the meeting.

This uses the exact structure you are learning for expressing time before an event: за + Accusative case for the amount of time (день) + до + Genitive case for the event itself (встречи). You can think of it literally as "at the distance of a day before the meeting."

Why use the short-form adjective занят instead of the normal занятый?

Short-form adjectives are often used as the predicate of a sentence to describe temporary states or conditions. Since he was busy at that specific time (a temporary state), занят is more natural here than the long-form занятый, which would imply a permanent characteristic (like being a generally "busy person").

Why is the perfective verb позвонил used with не here? Doesn't negation usually take the imperfective?

While the imperfective is often used with не to show that an action simply didn't happen in general, the perfective не позвонил is used here because it refers to a failure to complete one specific, expected result. He was expected to call at a specific time ("a day before"), and he failed to produce that result.

Why does the word for meeting (встреча) end in -и here?