What to do

I will be flying back to LAX from Narita Airport on June 2nd 12:00am.

I want to enjoy as much of Japan as I can even on the last day. Is it possible for me to check my luggage in early and still be able to explore the surrounding area? Is there anything nearby the airport?

You response will be greatly appreciated!

I assume you mean 12:00pm (noon), because I don't think Narita has any departure after around 9pm.

As for early checking in luggage, I think it depends on the airline. At least some airlines will not accept luggage check in until around 3 hours before check-in. Check with your airline about their policy.

In any case, I think you should stop sightseeing and be at the airport by 10am at the latest. That gives you some time to do sightseeing. You could either stay the last night in central Tokyo, do an early-morning activity and catch the airport train by around 9am or stay the last night near the airport and see an attraction in Narita.

Tsukiji Market could be an early-morning activity in Tokyo, but note the closing days, while the following page introduces some attractions around the airport:
http://www.japan-guide.com/e/e6400.html

Actually I mean January 2, 2013. It is acutally 12:05 AM, so it's a little past midnight.

My problem is the luggage. I don't want to have to book a hostel just to leave my luggage nor do I want to carry it around the entire day until my flight. Any suggestion to go about this?

Then, I am sure you must mean Haneda Airport, and not Narita Airport!

My problem is the luggage. I don't want to have to book a hostel just to leave my luggage nor do I want to carry it around the entire day until my flight. Any suggestion to go about this?

There are thousands of coin lockers in train stations across Tokyo. Just leave it in one of them.
http://www.japan-guide.com/e/e2274.html

Enjoy your trip!

haha yes it is Haneda! :]

Thank you so much!! :]