I would most likely think that it's just a cosplayer who just likes to dress-up outside of comi-cons and halloween.
But if I did find out that it really was my favourite anime character who somehow ended-up in my world by an unknown phenomenon, I would think that that was amazingly cool. But I would keep wondering how he got here, and how to get him back. I would also hangout with him, give him a place to stay and food.
I would try to trick Azusa into saying Nyan, like asking her if she was able to say an sentence that ended with nyan, as an "challenge". If I managed to, I would say "That's so... CUUUUUUUUUUTE!!!! *-*" and faint.
If I saw Ai, I would try to hug her, always wanted to do that. Also because as I don't hold many serious grudges at people, I think someone would have wanted to take vengeance at me, so one last hug before hell would be fitting.
Aand if I saw Alucard, I would be probably dying due to brutal dismembering right after it, so wouldn't be able to do much.
I would first make sure the character was real then capture them thus being the first person to discover an anime character in the real world i would then retire to an island somewhere with the money i gain from my discovery
Favorite male: It would be impossible to approach him from my viewpoint, but if there was a chance to sit down and talk, we'd talk about our ideals. There are times when I could think of wishing to join his cause, but other times I would stay the hell away.
Favorite female: Hard to approach, given how fixated she is with her dream. I think I could muster up the courage to ask for an autograph, but to talk with her face to face would prove difficult. Her life is to perform and I'm not close enough to her to cross that barrier.
Act cool and start a (rather ill-matched) bromance with either Cadis Etrama di Raizel or Vash the Stampede.
Sadly can't seem to pick a female character, lol. There are simply too many with various enjoyable/interesting personalities ^^
I guess I'd first freak out, then feel excited as hell before bracing myself and figure out how to make an impression from afar before trying to approach them in any way so I won't appear like a total douchebag or desperate stalker.