How does this new justice system work?
View the justice screen as usual, and you can scroll down through the various current crimes (and anyone who's been punished or is in line for punishment). Press enter while you're viewing a case that someone hasn't been punished for, yet, and you can select someone to punish. I think shelling "justice" out for people who don't deserve it might tick off your dwarves, though.
If you're lucky, the crime will have had a few eyewitnesses, which are noted in the crime's info, and anyone accused will be at the top of list of people you can choose to punish. Just be careful because sometimes the criminal himself will accuse someone. "Someone" in this case can even be an animal for now (giving you a good idea of who really did it), but that's supposed to be fixed in the next release.
I don't know how mandates not being met works, I'm assuming nobles will still choose someone to punish themselves rather than letting you do it, but I'm not sure. Other than that... hammerers are back in, and can be assigned from the start. I haven't had much luck getting them to actually give hammerstrikes though, my captain of the guard still usually seems to smack the criminal around. (I ended up un-designating my cages used as prisons, since that basically just seemed to make vampires I locked in them go out and drain more dwarves dry as soon as they got out, anyhow.)