How do you do dungeons with blu mage?

[Niwashi]

You must learn "1000 needles" first because you will never do enough damage to enemies and boss with just the water skill.
And learn enough skills (10) to get White Wind from the NPC near Martyn. It will help you heal if needed.

Well, the first required dungeon is Tam Tara. You have to do that one before you can do your level 20 quest, so you should really start trying it as soon as it opens up at level 16. But you'll want to be in your mid 20's or so before you take on the cactuars that can teach 1000 needles. So in at least that particular dungeon, it's reasonable to not have 1000 needles yet (though if some people do, it's nice). For all the higher ones past Tam Tara, you're probably right that 1000 needles becomes de facto required (and that they all require having significantly more than your starting water skill, even that first one).