Is it possible to do a day trip from Lucerne, Switzerland to Neuschwanstein Castle? Are there tour companies that will do this?

No---too far.

According to www.viamichelin.com, it's over 8 hours round trip.

You can hire a helicopter - it is the only way to do this trip.

Absolutely nobody in Switzerland is interested in Neuschwanstein castle.
Switzerland counts 1042 castles, palaces and fortresses (some of them in ruins).

As to the timetable, it would look like this:

Lucerne dp 6.10 - Fuessen ar 11.54
Fuessen dp 18.06 - Lucerne ar 23.49
Fuessen dp 19.05 - Lucerne ar 2.25am, Friday/Saturday and Saturday/Sunday only

Why go all that way to see a modern castle built by a crazy king in Germany versus a bunch of the real thing in Switzerland?

I have been to many Swiss castles and Neuschwanstein Castle, in the same trip, but I had 5 days in Bavaria, Germany and 9 days in Switzerland.
I suggest, you visit Gruyere Castle and Chillon castle. they will be quite rewarding.
Neuschwanstein Castle is not something to be seen on a flying visit. there are a lot of things around it too in Bavaria and Germany. make a separate trip to see it.
Have Fun...

from Lucerne to the castle directly for sure no but you have to change in Augsburg or Munich
in case you change in munich you have 2hours to get to the castle
wach this video show you how you get in the right train

How To Get To Neuschwanstein ( From Munich Main Station ) Ep 1
http://www.youtube.com/watch?v=wKY4Ymdfuog
How To Get To Neuchwanstein ( Füssen Station ) Ep 2
http://www.youtube.com/watch?v=L0_WFiXy4U4
for more info check my blog http://new-swan-stone.blogspot.de/ as i worked in the village i noted all info you may need