As an editor for a magazine, an assistant to a freelance editor, and a tutor of a writing center, I find it interesting how the critiquing dynamics dramatically shift between each. You approach chapter one of a novel much more differently than you approach a short story, and you certainly have to approach both differently if you're being forced to critique on the spot, just like a tutor must.
Well, there isn't a whole lot of advice for critiquing creative writing on the spot, so I'm going to dish out some tips from my minimal experience with tutoring. Writing centers usually receive a lot of technical writing papers, but there are the few rare who come in with creative writing, and while the tutors are helpful, many may not be creative writers and so don't know exactly how to approach a creative writing piece. Here are a few tips on how you, as a tutor, or even a beta reader forced to crit on the spot, can approach a piece of writing you aren't able to devote extensive time to:
1) Look at the hook. Does it open with character? Description? What? Short stories, especially, need to open with the character, because opening with description is often pointless. Setting description is often best saved for later, AFTER you've established the character. Stories are about characters, so they need to start with characters.
2) How long does it take for the story to start? In short stories, depending on the length, the story needs to start within the first page. "Scattershot," a short story I accepted to go into The Corner Club Press, is a 7,000 word short story whose plot immediately unfolds within the third paragraph: the MC hears a metallic ping, a piece of glass flies at him, and he discovers three holes in his window. So readers expect at this moment for the story to be about uncovering who did this. And this is exactly what the writer delivers, with a few exciting twists. Any short story, really, regardless of length, needs to start within the first page. Because they are so short, the mentality of reading a short story versus a novel differs. People expect more immediacy in a short story than they do a novel, so that immediacy must be delivered to maintain reader interest.
3) How exactly does the story start? With action? With dialogue? You often don't want to start a story with dialogue, as you want readers to care about the character first before they can care about what the character says. It CAN work, but there needs to be a reason it can't be saved until after you've introduced the story in a non-dialogue way.
4) Are you able to get the gist of the plot? Short stories only have one plot. Never really any sub-plots. If there are sub-plots in a short story, it's often going to be too short to draw those sub-plots in and create a satisfying close.
5) Is there any other way the writer can go about telling this same story? The job of an editor, I've learned, is to break the box and look beyond what's there. The job of a beta reader, I've also learned, is to look at what's already there and make it better. The job of a tutor, I think, should be to break the box and offer other ways of going about the same thing. Can this scene be introduced earlier? Is there a better way you can go about getting your MC from point A to point B and make it more interesting? Is there something else you can add to develop this particular scene?
6) What's the POV? How many main characters are there? The POV, whether first, second, third limited or omniscient, needs to be easy to follow. With third omniscient, the biggest mistake I see writers make is giving more time with one character than the other. The other mistake I see is introducing one character, building up the importance of that character, then deliberately switching POVs mid-scene without having established the importance of the character the POV is switching to.
These are just some of the ways to go about critiquing on the spot, and probably the only ways you'll really have time for. But these are essential points to hit upon. Addressing copy editing errors is important, but not as important as addressing these issues first.