Learn Italian: Anna mi ha assicurato che sarebbe stata puntuale. - Anna assured me that she would be punctual.

This is the "future in the past" rule in action! Because the main verb is in the past (ha assicurato), the second action that was in the future from Anna's perspective must use the past conditional in Italian. English simply uses "would" + verb, which makes this a very common trap for English speakers.

Why does stata end in "-a"?

sarebbe stata is the compound tense (past conditional) of the verb essere (to be). When a verb uses essere as its helping verb—and essere always uses itself!—the past participle must agree with the subject in gender and number. Since Anna is feminine singular, the participle becomes stata.

Is mi a direct or indirect object pronoun here?

It is an indirect object pronoun. In Italian, you assure something to someone (assicurare qualcosa a qualcuno). mi is the natural, unstressed way to say a me (to me).

Would the word puntuale change if Marco was the one assuring us?

No, it would still be puntuale. Adjectives that end in "-e" in the singular are the same for both masculine and feminine nouns. It would only change to puntuali if the subject were plural (for example, if Anna and Marco were both going to be punctual).