Learn Finnish: Saapuiko kirje perille tiistaihin mennessä? - Did the letter arrive at the destination by Tuesday?

The postposition mennessä, which means "by" a certain deadline, always requires the preceding word to be in the illative case. Therefore, tiistai (Tuesday) takes the -hin ending to become tiistaihin.

Since this lesson is also about asti and saakka, could I use them instead of mennessä here?

No, because it would completely change the meaning. Mennessä sets a deadline for a one-time event to be completed. Asti and saakka mean "until" and describe a continuous action. Because a single letter arriving is a one-time event, saying tiistaihin asti (arriving continuously until Tuesday) would sound very strange.

What exactly does perille mean, and do I need it in this sentence?

Perille means "to the destination" or "all the way there." It comes from the word perä (rear or end) in the allative ("onto") case. While you could technically just ask Saapuiko kirje? (Did the letter arrive?), Finnish commonly adds perille to emphasize that the subject successfully completed its entire journey.

Just as a refresher, how does the word saapuiko break down?

It is the verb saapua (to arrive) in the past tense (saapui) with the question clitic -ko attached to the end. In Finnish, yes/no questions are formed by attaching this clitic to the word you are asking about and moving it to the front of the sentence.