It must be completed by December 26th, but it's best to complete it by December 22nd for story purposes. (But don't worry about completing it extra early if you have S. Links or quests to work on, because the game will skip ahead to Christmas upon completing the Dungeon.

When can you spend time with Adachi?

In order to kick off the Adachi social link, wait for 13 May. On or after that date visit Junes and speak to Adachi - you'll be given the option to hang out with him. After some inane chatter that doesn't impact your s-link, you'll rank up to 1.

When should I do dungeons in Persona 4?

Complete dungeons as early as possible, with one exception

Generally in Persona games, it pays to get ahead of your deadlines. The earlier you complete a dungeon, the more time you have to work on your social links, as they'll be available more often after you've cleared that month's task.