I’m not looking to study abroad to earn degree credits, but rather I want to study abroad to gain international experience. The country I want to study abroad in is Japan since I majored in Japanese but never got the chance to study abroad there when I was an undergrad (I was concerned with finishing my major and graduation requirements so I could graduate in time). However a lot of programs I looked into seem to be geared towards either undergraduate or graduate students, and I’m neither. Is it still possible for a non-student like me to study abroad? Are sites like studyabroad.com and goabroad.com reliable?
Of course you can. Go! Sorry, can’t advise on the sites.
There are things like this for graduated individuals who want to learn a language: http://www.ef.edu/ils/destinations/japan/tokyo/
Otherwise your best bet in terms of an actual, structured program would be to teach English in Japan. Look into the JET Programme. Beyond that, you could simply go to Japan and live there on a tourist visa for a while. That’s not study abroad though; it’s just living abroad (and tourist visas don’t enable you to work).
I believe you can. Hm, have you checked on diversityabroad.com. I actually applied in their study abroad programs. 
hope it helps
Fulbright has programs for recent grads to teach abroad.