Your fastest option is the Shinkansen, the cheapest (but longer and more complicated) is a nightbus. The Shinkansen will be around 8500yen and the transportation into Osaka around 1000yen.
In Tokyo there is the JR Lines (Train)
http://www.jreast.co.jp/e/info/map_a4ol.pdf . They are a little bit cheaper than Subway and were my favourite. You will mostly stay inside the city center, the green line runs a loop and you can do alot of sightseeing with that.
The subway can be more convinient if going to places like Roppongi or the Tokyo Tower
http://www.tokyometro.jp/global/en/s...outemap_en.pdf It only runs in the center, so you can explore the whole map for sightseeing spots. There are 2 different subway companys, but both are on the same map
There are other private Trainlines but u will usually not need to use them.
Get a Suica RFID chipped card and you won´t have to worry about buying tickets. Just reload the card when its empty and pass the gates, the price will be calculated automaticly.
To find trainschedules and prices you can use this page
Hyperdia | Timetable and Route Search in Japan.
Have fun
Edit: Why do you have to apply for a one month visa? Most countrys get the Visa when entering the country. Just curious, but where are you from?
EDIT 2: Sorry, i did only post the fare, there is a seat charge of 5500yen for the Shinkansen. Hyperdia states a total price of 14050yen for the ride from Shinosaka to Tokyo