Okay. If it was me, I'd change at least half the water per day, and that's if I wasn't feeding-- remember, I'm thinking very short term here.
Feeding is more problematic with no filtration. A rudimentary way to set up a bit of filtration is to take a bit of used filter media from a running filter, whether poly floss, sponge, some bio-beads or whatever, then contain it in some way-- which is to say wrap it in a bio-bag, mesh bag, some cheesecloth, women's nylons, something that allows water to flow through-- and place it near some water flow, which you can create with a decent air bubbler. And/or-- put some sand or gravel from an established tank in the bottom of the bucket-- from a well established tank this will also have beneficial bacteria.
Believe it or not, this will create some nitrogen cycling in the bucket, especially if you create some current with an air pump. You might remember the derecho storm a few years ago that hit some eastern states-- in any case we had a 9 day power outage where I live, also a heat wave. I kept several tanks of fish alive, using a scaled up version of what I just described with battery operated air pumps. I tested, and ammonia stayed very low to undetectable.
Meth blue can be, but isn't necessarily a problem for beneficial bacteria. For mature tanks and filters not a problem at all ime-- I've thoroughly tested this. For a set up like I'm suggesting, not sure, so I'd keep an eye on ammonia in the bucket.
All of this would be minimal and short term like I mentioned above. Again, I'd recommend either setting up a tub for a few bucks or, better yet, set up a 5 or 10 gal desk top tank with a small, cheap filter, if this is going to last more than a few days before getting him into the main tank with a divider. In my case I'd do a tub over a bucket in the first place-- more surface area to water depth means better oxygen exchange with the atmosphere, better than a bucket-- but in the bucket scenario it would only be an emergency option until I see he's revived, looks like he has a shot at surviving the ordeal, and then I'd get him into better living quarters one way or another.
Feeding-- again, no filter in the bucket complicates this, besides the energy/stress issue that means he needs to recover some before eating much. SO-- if he is willing to eat, I'd feed very lightly until he perks up a little bit, just a few flakes of flake food per day, by a few I mean two or three or four, etc, not a spoonful. Alternately a small freeze dried shrimp or two, just enough to provide a little food energy without requiring much energy for digestion. No pellet or sinking food. That way if he doesn't go for the food fairly quickly you can net it our pretty easily with minimal effect on water quality. Again, if he keeps improving this would be as short term as the bucket should be.
