If polycarbonate plastic is a polar material with ester bond linkages which usually hydrolyze in acids, why is it resistant to both weak and strong acids (as polar substances)?
Looking at what has been published about hydrolysis of polycarbonate resins to recover bisphenol A, typical conditions involve water, heat, optional base catalyst (or more rarely, acid, which probably further degrades BPA), and very often a cosolvent such as tetrahydrofuran that itself is stable to hydrolysis. So your intuitions about whether polycarbonate should be susceptible to acid hydrolysis are correct.
There has been a lot of attention paid to whether polycarbonate in household items (e g. baby bottles) releases bisphenol A into foods, causing health concerns- which it certainly can.
On the practical side, such as preparing dilute aqueous acids or bases in a polycarbonate container at ambient temperature (e.g.for lab use), degradation would be very slow due to the density, hydrophobicity, and perhaps cross-linking of polymer chains into a tight matrix. This prevents the aqueous acid or base from physically contacting the bulk polymer, although slow surface degradation is still likely.