Approval voting is an unforced preference (i.e., allows ties) ranked voting system with only two ranks just like bullet voting systems (plurality and majority-runoff), differing in that it allows ties in both ranks and not just the second.
Arrow’s Theorem applies to unforced preference ranked voting systems (with or without limited numbers of preference ranks) the same way as it does to forced preference ranked systems.
Perhaps surprisingly, no, approval voting is actually a score voting system, which provides more information than a strict preference ranking.
This isn't necessarily intuitive, and our immediate impulse might be to object that 2^n-1 is less than n!, but as a quick informal illustration, note that the voting system where each voter assigns each candidate a score in [0,1] and selects the candidate who earns the highest sum from all voters fairly straightforwardly violates Arrow's theorem. Now, consider an approval vote where each voter rolls a single random number in the range [0,1) and vote for each candidate whose score exceeds their random number, and we get asymptotically the same result (but with some error bars). It turns out that scoring, even on a 2-point scale, is just a better primitive operation than ranking!
(Also, it helps that 2^n-1>n! for n=2,3, which covers a surprisingly large proportion of interesting elections.)
Arrow’s Theorem applies to unforced preference ranked voting systems (with or without limited numbers of preference ranks) the same way as it does to forced preference ranked systems.