Though there are several species of tube anemones the most common and like the one you describe Cerianthus sp. is primarily noctournal and very easy to care for so it should be a good one for you. There are a few cautions to keep in mind: Generally these things pack a pretty strong punch as far as stinging goes. While mine never nailed any fish for feeod he did give my tang a bit of a scar (which healed nicely) but most fish will avoid it anyway. Also since it is a creature of the night it will expand to it's max in darkness. It is non-photosenthetic so light is not a must. I fed frozen silverside minnows about once a week. It will grow quickly if well fed.
As far as illness is concerned or it not eating; was the "tube" around it foot in good shape? It should have looked like it was covered in sand. Also where do you have it placed? It should be in a crevice of rick on the substrate and after a few days will bury it foot to the point where you would have to dig it out. It it's foot is not set yet this may be a reason it is not eating.