So it's not really me killing this plant but it's not looking good. It was a gift to our office and no one is taking very good care of it. My track record with plants is not good either. Anyway thank you in advance. http://img.villagephotos.com/p/2005-11/1104980/tree.jpg
It might not be as bad as you think. It probably needs to be transplanted into a larger container, but start by making sure it is in a place where it can get some natural light. Can you get it near a window? It may need some extra humidity. Small mist bottle every few days would help. Heaters dry out the air, and this is a tropical plant probably grown in a greenhouse. Go ahead and trim off those browning leaves near the trunk. Looks like it has a good chance. Water it regularly but don't drown it. Good luck, Someone else will have some more advice on plant food.
I agree, the photo of the palm doesn't look that bad at all. It could be an Areca palm "Dypsis lutescens" common name (butterfly palm) The pot size looks ok to me, an inch or two wider than the rootball is all you really want to go. Try and give it good bright indirect light and water it when the top 2" feels almost dry but still a little bit moist. You don't want it wet or to dry out. I'd take it out of the basket and water it over a sink so the water drains out, or get a saucer for underneath and use a turkey baster to empty it so the plants not sitting in water. As far as fertilizer, spring and summer months 1/3 strength once a month is fine. Hardwater can cause some of the fronds (leaves) to get brown tips. Water that sits out like a plastic jug minus the top for 24-48 hrs evaporated chlorine which can cause the browning, unfortunately evaporation won't happen with fluoride, another reason for browning. Bottled water is another option. If it's watered correctly and has the right light, the brown tips won't be that bad and won't effect all the fronds. Fronds that look really unattractive can be snipped off. If cared for right, your palm can reach 10' or higher, but don't worry, that can take a quite a few years.....