Where can I find a good mp3 recording of some of the tunes (like a
ringtone) they play in Tokyo, Japan, when a subway arrives at the
station? (There are differnt tones for differnt subway lines.)

Dear aweigend-ga,

When I saw your question I must honest and thought: not much chance
finding them. Well, I am amazed. It seems that it is almost a hobby in
Japan to go around recording all the tunes used on the metro and
railway stations.

Some of the sites are in Japanese, so you if you are not familiar with
the language, here is some advice to find the files (and pardon me if
you know this already). Most of the files are mp3, wav or wma. You
need to drill down in all of the sites to reach a particular station.
There you will find the link to the tune. When you hover over a link,
look in the bottom left corner of your browser and you will see the
name of the link. If it says html at the end then that is to another
page and you can click on it. If it ends with mp3, wav or wma then
that is the file you are seeking.

Recordings are of varying quality.

This site has some departure tunes for the Tokyo Metro.
Namboku Line
http://hisaai.at.infoseek.co.jp/Eidan/07_s_eg.html
Mita Line
http://hisaai.at.infoseek.co.jp/Toei/06_s_eg.html
Tojo Line (1 example) 
http://hisaai.at.infoseek.co.jp/Tobu/03_s_eg.html


On this site for the Tokyo Metro drill down to the station until you
see an image stating ?ON? (white on black). Click this to access the
mp3 file)
http://rokko1go.jpn.org/main/oto/oto_tetu1.htm


The next sites are for the Japanese Railways.

Japanese Trains - Drill down through the links to the station and look
for the link mp3 or wav or wma
http://melody.pos.to/page/jr.htm

Numerous Japanese railway examples
Well labelled: wav mp3
http://www.hatsumelo.com/line/

Departure Melody web site. Drill down through the links to the station
and look for the link mp3, wav or wma
http://hazami.jog.buttobi.net/

Shinkansen bullet trains
http://www.japan-railways.com/musice.htm
http://www.japan-railways.com/yatej.htm


I hope this answers your question. If it does not, or the answer is
unclear, then please ask for clarification of this research before
rating the answer. I shall respond to the clarification request as
soon as I receive it.
Thank you
answerfinder

Nicely done, answerfinder!  Such a fun answer!