If your main character is Caucasian, then she would call the new girl whatever the Asians have been calling them, concerning race.
If your MC is NOT Caucasian, and doesn't know the word, then typically what people have done historically is pick some prominent different physical feature about a new race and use that. "Negro" is just "black" in other languages. Your character, compared to Asians, may have brown or yellow hair, or curly hair, a thin face, light pink skin, a large bust. Blue or green eyes. A wrinkle in her eye lids.
Your MC should use the language those around them use. If in an interaction with the new character, the new character reveals she is "Caucasian", the MC can use that thereafter, but would more likely use the girl's name!
As to whether the narrator uses "Caucasian", I personally would not. The way I write (3rd person limited) my narrator may understand my MC's emotions better than the MC herself does, and my narrator is more eloquent in description than the MC, but my narrator doesn't know anything the MC does not know.
So if I were writing, my narrator may describe the girl as seen by my MC, three or four outstanding differences my MC would quickly notice about the new girl (with fresh language to avoid scientific or racist terms), and then my MC would pick one of these prominent features, or if she was Caucasian the feature she herself was known by, and use that.
"Oh my gosh," Alice said. "You're another blue!"
The new girl looked confused. "What's a blue?"
Alice pointed to her right eye, with a grin. "Our eyes are blue! I'm Alice, but everybody calls me Blue, after my eyes, and now there are two of us!"
"Oh. I'm from Norway. Very far north. You're probably from the far north, too, blue eyes are very common."