It depends on the bullet in question. In TSR, Mao takes a bullet to the shoulder from a pistol, but there's no bleeding (that I can remember, anyhow), though that was from a pistol.
Most Classes of body armor can stop pistol rounds, but only Class IIIA and Class IV body armor are rated for stopping rifle rounds. Standard rifle rounds would penetrate most forms of body armor, nevermind armor-piercing rounds, which is why the US and most NATO countries use a kevlar "soft vest" with hard armor plates (ceramic, in the US' case) inside the vest, for added protection. It's unknown if the AS suits hold trauma plates inside, though, but it seems unlikely.
What was he shot with? Judging from that pic, I'd say either a rifle or an SMG firing armor piercing rounds.
Definately wasn't shot with a Barret M82

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Fire-resistant is definately plausible; current Nomex flightsuits are fire-resistant.