The particular pitcher plant you have there is a
Saracenia rubra.
First, get it out of the death cube. Death cube + green house/outdoors in any sun = probably way too hot for that plant. It will adapt to outside humidity with little difficulty.
Never ever pour water into the pitchers. It won't be able to digest food properly and may topple over from the weight. It can handle a little splash-in from heavy rain, but those hoods are there to prevent water from entering the pitchers. If a bug enters a pitcher and that pitcher has no fluid in it, the pitcher will detect the food and make fluid on its own. Adding water dilutes the digestive enzymes, rendering the pitcher useless.
Second, it actually looks healthy. Individual pitchers die all the time; they grow new ones. Third, the plant is probably not sun-hardened, therefore all the pitchers may dry up and shrivel away within a day or two - this is expected, so don't sweat it if that happens. Keep caring for it like any healthy sarracenia, and it'll bounce back this year or next (depending on when it goes into dormancy for the winter), and you'll start seeing new pitchers. I have a rubra from a local nursery, not even as healthy as yours, and all the pitchers were dead in less than a day in full sun - it now has many healthy new pitchers, a month or two later.
I recommend the following:
1. Remove from death cube (use scissors to cut away that flappy plastic piece if there is one, don't try to squeeze the plant through the hole). Don't repot - its current medium is sphagnum peat moss with a top and/or bottom layer of dead long-fiber sphagnum, which is perfectly fine for that plant. Repotting now will only shock it and set it back further - repot it next March or April, 2011 (early Spring).
2. Once out of the death cube, put it in a 1-inch deep tray (plastic only; ceramic may leech unhealthy amounts of minerals into the water/soil) and keep
distilled or
rain water in it. Don't use tap water or well water, and don't use "purified drinking water" or "spring water". Any water used should be less than ~50ppm (distilled is 0-5, rain water is usually around that range or a little higher). The width of the tray isn't of high importance, but I like to have my water trays about 2-inches wider in diameter than the bottom of the pot so I can add water easily, and so it doesn't evaporate as quickly.
3. Rubra like wet soil, so keep the tray filled as often as possible. It's fine to let the tray dry out, but never let the soil dry out. Personally, I almost always refill the water tray before it dries out.
4. If it isn't out in the rain, top-water the soil at least once every two weeks to oxygenate the soil.
5. Six (6) hours or more sunlight per day is ideal, the more the better.
Good luck!