Learn Russian: Старшая сестра работает.

Старшая is an adjective describing the feminine noun сестра (sister). In Russian, adjectives must match the gender of the noun they describe, and -ая is the standard ending for feminine adjectives in the nominative (subject) case.

Is старшая the exact same word as "old"?

It comes from the word for old (старый), but старшая specifically means "older" or "eldest". In Russian, you use this specific form to mean "senior" or "elder" when talking about siblings.

Why does the verb работает end in -ет?

The dictionary form of the verb "to work" is работать. Because the subject is "sister" (which is "she"), we drop the -ть and add -ет to show that one person is doing the action in the present tense.

Where is the word for "the" in this sentence?

Russian does not have articles like "a" or "the". You just say "Elder sister works," and the listener understands from context that it means "The elder sister."