As with MT, there is an install directory and a blog directory, which is where your blog is rooted. If desired, these directories can be the same, but I prefer to keep them separate.
By the way, did you used to build your MT blog into tokyotimes.org/mt/ before? When I go to that directory, I see your blog, but only up through September 2004.
So, you can definitely install WordPress into tokyotimes.org/wordpress/ without an issue. You should even be able to use the tokyotimes directory as your WP blog directory.
MT creates an index.html file in its blog directory. WP creates an index.php file in its blog directory. The Apache web server is normally considered to search for a default main page in a specific order. Usually, it will default to index.html before index.php.
So, you could point your new WP blog at your root directory, and test it out by directly accessing http://tokyotimes.org/index.php. When you are ready to switch over, you could rename index.html to something else, like mt.html.
Which brings me to another issue. I'm assuming you will want to migrate the content over from your MT blog to your WP blog. I blogged some notes on how I did the migration.