I understand that Anniversary has dead ends and other outdated mechanics and I don't wanna deal with it. Is it necessary to play it first to understand this one? Or are they stand alone in terms of plot?

In case you're still wondering:
Anniversary's last mission is storywise loosely connected to Judgment Rite's first mission.
But you won't face any problems if you haven't played Anniversary.
Apart from that there are no other overlaps between the two games' missions.