I'm feeling a bit lazy so I'm going to utilize bullet-points.
1. So essentially I think this confirms that Saber is Link until proven otherwise. So that's nice. Could be misdirection, but I doubt it.
2. In terms of Archer, I'm still really unsure. The last bit seems to imply that she's Japanese, though that could just be what Akira wants her to call him so in the long run it doesn't mean much of anything, yet.
3. Also Akira and Rei have some connection, so that's interesting. My guess is that Akira has a crush on Rei, but we'll see.
Now in terms of critique:
1. There are a few typos, but nothing major and I could still understand what you were trying to say. The writing is solid from a technical standpoint.
2. There's a couple places where the structure of paragraphs is a bit weird- the biggest example of this is at the beginning of the chapter-
Rei poked at her breakfast. Normally, being a morning person, she would have finished by now. However, the events of the previous night had left her exhausted... A jellied mass, it somewhat resembled clear tofu or a block of pudding
I feel like the last chunk should be after "Rei poked at her breakfast." It sounds weird in its current location. Need to make sure that the ideas in paragraphs logically follow from one to the next.
3. I think it might be a good idea to actually tell people what happened during World War III. If it's going to be related to some future twist, then obviously don't. But if not, it'd give readers a good overall context for what's going on and a better feel for the current world. We're be able to see WHY weather got all screwed up- I'm expecting fallout from nuclear weaponry, but we can't be sure without you telling us. You seem to want to do worldbuilding, and that's good (and the worldbuilding you've done so far is fine), but I think a very important part of the worldbuilding is really explaining why this world is the way it is.
4. Although I get you kind of want to do a snarky narrator, but I feel like if you want to do that, you need to do it consistently. That doesn't mean you need a quip at the end of every line or anything, but having just one or two quips in several pages just makes the snarkiness distracting rather than amusing. It should be frequently present, at a minimum. It'd also be beneficial for there to be some identity to the writer that explains why he's telling the story in such a snarky way, but that isn't necessary.
5. In terms of characterization, you're doing pretty well. Rei is a solid protag- I'm not in love with her yet, but she's good. Saber... is ok. Nothing great- he's kind of entertaining at times but outside his repartee I'm not sure he's particularly well-defined as a character. Maybe fleshing him out a bit more would be a good idea? The others I still don't really know enough about to judge.
6. You're doing a good job at adding description and descriptive language into the story- I'm capable of picturing everything and really following along, so that's good.
7. One other little thing is that I'm not sure why Rei's dad would have clothing that were proper early last millennium in his closet. But hey, we don't know him so maybe he's just into that style of fashion or something.
Sadly, exams have eaten up my writing time...