I saw on a documentary that it is physically possible to get a third of the way up the Eiger through a chute that starts from an underground train line.
Does anyone know about this? Do they let tourists up there? If so, how much does it cost?

There are 2 stations inside the Eiger. Eigerwand and Eismeer.
panoramic map (10 MB)
Their homepage lists however only one hiking trail starting from Jungfraujoch (the end station).
Than there is a trail at the foot of the Eiger North Wall.
I you are a well equipped and experienced mountain guide, you shouldn't have too many problems.
Tourists are admitted to the Eigerwand station, which is near the door that opens on the wall -- all trains stop there for a few minutes (Wand=wall, in case you were wondering). That opening does exist (remember the Eiger Sanction movie?), but you can't and shouldn't think of using it, as indicated by #2.

The climb up is a bitch but the ride down is easy. Take a look. http://www.mefeedia.com/entry/speed-riding-down-the-eiger/11809435/

The swiss television made once a kind of docudrama about a climbing the Eiger Northwall.
They had to wait more than a year to get the right weather.
With modern equipment is not that difficult anymore like in the beginning.
But you can't stop the weather and the avalanche with modern equipment.
The Eiger Northwall is only for people who know exactly what they are doing.