I used to have a Walkman set up in my car. I started out with an 1/8" stereo to RCA adapter into an amp as mentioned before.
To solve the battery issue, I made a simple voltage regulator to replace the batteries. I used a CB mic cord (5 leads) to connect the walkman. 3 leads for the sound and 2 for power. I installed an 1/8" jack at the end and pulled the 2 extra wire out the top and down the side of the radio into the battery cover. The other end ran under the seat to the voltage regulator and amp connections.
Worked great. The radio reception sucks though. I'm sure there is a way to plug in an mp3 player in a similar fashion.
Or, you could try one of these...
SecretAudio
I think its a bit expensive.