Can an outsider study Japanese martial arts in Japan?

Indeed, there you will find many dojo that teach various systems (Karate-Do, Judo, Iaido, Kendo, Jujitsu, Kenjutsu, Ninjitsu, Aikido) and styles (Shorin Ryu Karate, Goju Ryu Karate, Shotokan Karate, Wado Ryu Karate, Kyokushin Kai, Yoshinkai Aikido) and many non-Japanese Martial Arts like Kung Fu, Escrima/Arnis/Kali, Silat, Mixed martial Arts, etc. The question really is what art/discipline will suit you the most. I recommend asking more refined questions, and that way you may be able to discover what art might be more suited for your personal interests Sport, Discipline, Defense, Mind-Body Connection, etc.

It is always a good idea to get introduced to the dojo through contacts or instructors. Showing up at the door unannounced is never a good idea. Most are very generous with those that show the right courtesy and respect.

Also the Japanese were very wise by bringing forms of martial arts from China and Okinawa.