Well, one of my first things I'd want to know would be how she knows about me.
Going by the phrase "I'm Yumemi Okazaki, I've been looking for you, my husbando.", she'd know me as a character from some manga, anime or videogame whom she fell in love with. I don't know about you guys, but I for one would be curious about that manga/game/etc. (in my case, I think it'd be some obscure visual novel or a quirky Yume Nikki-like game).
But as far as our first RL meeting would go - no way I'd have to ask who she is. I'd immediately recognize Yumemi. I'd blurt out her name the moment I'd open the door. Sure, I propably wouldn't believe my eyes, and I'd still be flabbergasted and wondering what the hell's going on even as I feel her hug me and hear her say "I knew you'd recognize me!". But deep down, I'd know it's her right away.
And I don't think Yumemi would have a lot of doubts, either. I mean, she may be unorthodox for her belief in magic, but she's still a scientist and a professor. She would have checked and double checked wether she's in the right universe & timeline, where I live, wether or not this version of me really is in love with her, and when would be the best time to pay me a visit and meet for the very first time. She may be a little brash, but she's still focused and methodical.
I'd let her in, I'd want to ask her all sorts of things (though she'd propably beat me to it and end up starting the conversation – she'd invariably be the proactive one in our relationship, anyway), and we'd both propably feel a little awkward about actually being in the same room with one another. Heh, I can actually see myself fret about how I'm gonna explain this to my parents while she'd just walk up to them and introduce herself as my girlfriend. The excuse “we know eachother from university” would propably work, though I'd hope that she wouldn't insist on moving in with me...
I'd propably also wonder how she got to my world in the first place – but then again, this scenario would actually make sense with Yumemi. She already travels to other worlds/timelines/universes via the propability hyperspace vessel in canon, so this would just be her usual routine with a different destination.
That also puts >>416 & >>418 in an interesting light, btw. Someone like, say, Konata or Osaka logically wouldn't be able to bring anyone to the 2D world – but what about dimension travelling or supernatural waifus like Yumemi or Yukari Yakumo?
Not that it'd matter much in my case – Yumemi's own world was never actually shown in canon, Gensokyo is a little too crazy and dangerous for my tastes, and I actually like being in this world – but it's an interesting question nonetheless.