To simply say: There is no way you can get the password of the user logged on.
I tried the same thing before and was put down with no way of getting up.
As everyone else has said, $ENV{'REMOTE_USER'} contains username.
From chromatic's post, I must say I've never heard of $ENV{'REQUEST_USER'}.
So, no, there is no way to receive the password of the user from headers.
I hope you didn't really need the password, adn if you do, I have no idea!