Just wanted to mention for anyone coming across this thread in the future that there actually is a really good, easy way to do this without even having to install WordPress on the subdomain - all you have to do is point the subdomain to the same location as the main site and then follow these steps:
Step 1 is to enable "multisite" or "network" capability on your existing WordPress site. This is all built in to WordPress and fairly easy to set up following the WP documentation here: http://codex.wordpress.org/Create_A_Network. You'll want to choose the "subdomain" option, not the "subdirectory" option.
Step 2 is to install the free plugin "NS Cloner" which we built for exactly this kind of scenario from http://wordpress.org/plugins/ns-cloner-site-copier/.
Step 3 is to go to the NS Cloner admin page. Following the instructions there, you'll be able to really easily create an exact, perfect clone of your existing site at the subdomain URL with just one click. You'd then have be able to go to for example mysite.com and jp.mysite.com and see the EXACT same thing - content, widgets, menus, etc.
Step 4 is go to jp.mysite.com/wp-admin - you can now customize it and change language, images etc exactly how you need to without affecting the main site.
Pretty cool, no? Hope it saves you some time!