The voting system you describe is called Single transferrable vote (STV).
Arrow's theorem does not discuss STV specifically. Instead, it states that no electoral system exist that satisfies a certain number of criteria at once.
One of these criteria is monotonicity, and there are studies that conclude STV is non-monotonic.
In two words, about monotonicity.
Fix a certain voting profile (a set of 3+ Candidates, 2+ Voters, and each Voter has already voted).
Now, imagine that during the voting, Alice the Voter gave CandidateX a certain Rank
n (in other words, CandidateX is Alice's
Now, if the situation is possible that when Alice lowers the X's Rank, and after the votes recount, X gets better outcome from elections, then the voting system is non-monotonic.
The same is for the opposite situation: Alice raises their Rank for CandidateX, and this results for worse voting outcome for X.
There are several articles available that describe this phenomenon:
This opens the way to various techniques of tactical voting; in some corner cases, not voting for your preferred candidate improves their result.
Also, Monotonicity is not the only criterion that STV apparently fails to satisfy.
Good news is that Arrow's theorem does not "invalidate" STV because it states that there is no "perfect" voting system. Or, it does "invalidate" all voting systems, if you will.