Yea just a small spot of super glue on the edge piece of the cap, the par that fits down into the piece. It's a common problem, I just took all of mine out and glued each cap on to each piece. Just use a tiny amount, you don't want the glue to drip down into the cube itself at all.
Stickering's easy, you can use the applicator sheet, if it came with one. It'll be the large clear sticker sheet. just take it off and apply the sticky side of it to the colored faces of each sticker sheet. Lift it up and it'll have all the stickers stuck to it. Then take that over to the puzzle and apply all the stickers at once. Be sure to line it up good though or the whole side will be crooked. I prefer to sticker my puzzles one sticker at a time. I use a clean razor blade, and put just a tiny point of it under the sticker, lift it off the sheet and then apply it to the puzzle.