Here's what I got from the details, at least this is how I read that.
1. Chapter 1 Character Quests
Shirou, Bathym and Oniwaka planned on 8 March 2017.
Kengo, Moritaka, Garmr, Taurus Mask, Horkeukamui, Nomad, Ikutoshi, Magan, Claude, Snow, Pollux, and Andvari planned as well on a later date.
You need any one of that character's card (whatever variant), had that limit broken for the first time, and had a certain part of the main story quest cleared (e.g. Shirou's require you to finish Main Story Chapter 2) before you can play that character's quest.
(I really hope they don't necessarily require the limited-time variants. I only got Shirou and Gabriel for Valentine's. FML)
2. Limited-Time Quests and Additional Daily Quests.
There's already 3 new quests to get more coins in the Event Area of the map. They took place in Tokyo Casino.
There's also another limited quest that makes limit-breaking cards for the first time easier. (Helps you unlock those Chara Quests)
Limited Time Quests are from 3 - 20 March.
Also, about 2 more daily quests, elemental-related and coin-related starting from 4 March.