Crops and other products only wither while they're on your field/in your town.
Once in your storage, it'll stay there forever. I probably have at least 500 items in storage right now, myself. =w= I am a total hoarder.
As for specific whither times, I'm afraid I'm not sure. Sunflowers seem to stay good for at least 8 hours or so after they've finished growing, though. I usually plant them overnight, or when I'm heading to work, and they're ready to harvest when I log in next. :3