During the last inspection I did, I noticed a lack of eggs and larvae throughout the hive. Additionally, I saw 3-4 capped queen cells (one was toward the middle of the frame, the others were down low on the same frame). So I'm pretty sure my queen died and the bees have made arrangements for their new monarch to hatch in a day or so.
My concern is keeping the hive population from falling while the new queen is getting ready to start laying eggs, which could be a good 2 weeks or so out. This hive has always been a bit on the weak side and has developed slowly. I have another hive that is very strong, so I was thinking of taking a frame or two of capped brood from there over to this queen-less one. Is the timing of doing that particularly critical? Can I do that any time after the queen has hatched? Is there a better way?
Do it any time. If they are already weak, the sooner the better. But, I don't think it matters if you do it before or after the new queen hatches, so long as you don't accidentally transfer the queen.
First of all, it will take a good month if you plan to raise a queen from brood. I personally am not a fan of letting the bees raise an emergency queen as you are at an increased risk of getting a poor queen, who will inevitably fail late in the Fall/Winter. But regardless of where/how you get a replacement queen, I would recommend the double screen board method. You will get the advantage of 2 queens rearing brood for a short time.
http://robo.bushkillfarms.com/beekeeping/queen-introduction/