[Firebird]

There are 2 train going from the Airport to Ueno. One will be around $20 and will be a bit faster and the other one is a local train wich stopps everywhere and cost around $10.
From Ueno to Tokyo its only a short train ride with the Yamanote or the Keihei Line (4 stations) so it´ll cost something around $3, but i think even less.

The Alien Registration Card is called Gaikokujin Shomesho (im not sure how to write it) but its something like this in Japanese 外国人登録.
To get that you will need to find out where the ward office for your House will be (ask your company or the ppl you rent your room from). Im not sure how that other document is called and im also not sure if you can open a bankaccount with it, but give it a try. It is like a certification that you aplied for the Alien certification.

Edit: It took me 80 min from Narita to Ueno and cost 1000yen. Im not sure if there isnt a train running directly to Tokyo Eki, just ask at the desk.