Isn't "Oruta" suppose to be a pun on "Alter"? Since she's Okita Alter?
Yep! It's Okita (沖田) going to Okita (おき太) going to Alter (おる太); as this doesn't work very well in English, an equivalent one is being used for effect.
Though with Emiya getting 弓 (Bow), which is also shorthand for his class, it's also just being used as Alter(ego).
Updated
I'm not even the slightest bit sleepy...Erm...Coffee is amazing.Yes, indeed.
Coffee is...kindaamazing!!As in one of her NP lines, where she forgets its own name.Okitan
It's Oruta, with the ta being 太 - Okita is sometimes called おき太 as well.There should bea word for this surprisingness.Majin-san, wanting to vocalise her surprise.