Learn Russian: Наш переводчик работает очень хорошо.

Наш (our) must match the gender of the noun it describes. Because переводчик (translator) ends in a consonant, it is a masculine noun, so we use the masculine form наш. If we were talking about a female translator (переводчица), we would use the feminine form: наша переводчица.

How do we know to use the ending -ет on the verb работает?

The dictionary form of the verb is работать (to work). It follows the standard regular conjugation pattern. Since the subject is переводчик (a "he" or 3rd-person singular), we drop the -ть from the infinitive and add -ет.

Is хорошо the same word used for "good"?

Хорошо is an adverb, so it means "well" when describing how an action is done (like working). If you wanted to describe the translator himself as "good", you would use the adjective form that matches his masculine gender: хороший переводчик.

Why isn't the instrumental case used in this sentence?