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Is there such a thing as a one month railpass for Tokyo, and is it worth the price if I take the train an average 5 times a day?
No, as far as I know there are only commuter tickets, which are valid only for a specific route. Note also that there are multiple companies that operate public transportation in Tokyo, and each has their own tickets; there are tickets valid for multiple or all operators, but it's rather intransparent. PASMO or SUICA seem to be the most convenient. If you in fact use the train 5 times a day, then 1-day passes like the Tokyo Combination ticket or the 3-day Tokyo Subway Ticket (for tourists) may be worth it, but pay attention to what operators they are valid for!
http://www.tokyometro.jp/en/ticket/value/1day/index.html
http://www.tokyometro.jp/en/ticket/value/travel/index.html
Alternatively, can I rent a bike for a month, and if yes, how viable is it opposed to public transport?
I think it's an awesome way to get to know the city, and central Tokyo is pretty compact. Look at a map to see where you'll want to go and get a feeling for the distances.
I'm worried that August would be too hot to cycle for longer periods.
That depends on your fitness and tolerance for heat and sweat. But yes, August tends to be hot and humid.
Upvote:1
A one-month pass from Warabi to Suidobashi is 9050 yen, or 300 yen/day. You can get off anywhere in the middle (Akihabara, Ueno) for free and if you ride farther (Ginza/Shinbashi) the remaining fare is calculated as if you entered at the nearest station (to your destination).
A bonus hint: getting to Big Site by train is expensive. There is some bus service, which is cheaper -- basically from Shinagawa, Hamamatsucho, and Tokyo stations if I remember correctly. It's also possible to walk the Rainbow Bridge, but not bike across.