Japan One Month Study Abroad. What To Do?

I am heading to Japan with my university next summer and I am looking for things to do while I am there. Of course there is tons to do but I would really like to experience Japan as a native. I am doing a few tourist things with my University such as visiting a temple, going to Skytree and going to Disneyland as well. I am looking for things that I can do as a native such as places to visit and shops to go to. I'll also be legal drinking age when I'm there as well so I'm looking for a few fun places to go with my friends that would really help me experience Japan! It may be a broad question and I do apologize about that but as I am there for only a month I am trying to cram as much as I can into that month to feel like I am a native and see it as a native does. I would like to keep around Tokyo as much as I can as I only really have one full free weekend to possibly go and visit Osaka or something. I am on a tight budget as well so I am looking to do this as wallet-friendly as I can.

My other quick question is would it be worth getting the AT&T international plan for my phone or would it be cheaper to get a phone in Japan. Silly question but a telephone is important for connection with my travel group.

Thanks in advance for your answers!

Well... a native wouldn't be trying to cram as much as they can in a month, for starters ;)

But one thing I like to do that I might suggest, and doesn't cost any coin as all, is "urban hiking". Basically just walk the city, not as a way to get from point A to point B (though of course that's what the natives are doing...) but to see it outside of the stereotypical tourist hot spots. Tokyo is very conducive to this, because you can't inadvertently wander into a dangerous neighborhood (there are none.) Just get off at some random subway stop, and walk.

As for phones... I don't know what ATT's international plans are like, so I can't say what would be cheaper, but note that as a non-resident you can't get voice service in Japan, only data (though that may be all you need/want.)

sounds like an awesome study month.

being there a month, you should catch on to things that are native. you lifestyle and food habits will change.

I think the shopping and nightlife spots you should be able to see fairly easily over the course of your 4 week stay. ie - ikebukuro, shinjuku, harajuku, shibuya, asakusa, odaiba, etc

for a suggestion for your full weekend free. try to get out of the city. experience something away from the concrete jungles. but being only a weekend, it still needs to be close to Tokyo. Honestly if you wanna experience Japanese only type places. I'd suggest going to Hakone region. Baths, funky zigzag railway, castle at Odawara etc.

So pleased to read your question as a local :-)

As long as you have no problem relegiously and you will be in July or August, I would like you see Obon.
http://ishijls.blog.fc2.com/blog-entry-496.html
There is a program that you can see them.
http://ishijls.blog.fc2.com/blog-entry-491.html

Not only Hakone, but also Nikko is worth to visit for the weekend.