In MTG Arena, does your "rank" affect what cards you can draft?
Not at all.

The cards you're shown in a draft on MTG Arena are a simulated version of what you would see in a paper draft of MTG cards.

This means that, for each pack, with some exceptions, there will always be 10 commons, 3 uncommons, and 1 rare/mythic rare.

As the draft goes around, simulated players (these would be real players in a paper draft) pick cards, and so the next packs that you see will contain less cards.

If you need more of an explanation, see: https://mtg.gamepedia.com/Booster_Draft