Their bass has a classic punk sound that reminds me of blink-182. What type of bass guitar/amp/settings would one use to get this sound? You can hear the bass part semi-isolated at :40
The recipe for classic punk tone is a P bass, played with a pick, into an SVT. SVTs are heavy and not cheap, so thankfully there's a lot of pedals that do the SVT thing well - Master Effects Super 300, Catalinbread SFT, Broughton SV-Pre, lots more suggestions in this thread.