As others have said, I also think this is a totally realistic amount. Living on £1,200 per week is definitely more than possible.
Also, as everyone else has said, a month in Tokyo is really too long and you will get so much more out of your holiday if you travel around the country. However that means that your accommodation will likely work out more expensive as you are not getting a long term discount.
A PP mentioned Lonely Planet and I agree with that. Coincidentally, they had an article about two months ago on just this thing:
https://www.lonelyplanet.com/articles/train-travel-trip-japan-planning-guide
The Japan Rail Pass has gone up a lot in price but still makes sense if you are making multiple journeys during the period. A one week pass covering the entire country costs £270 per adult and a two week pass costs £430 per adult. Children under 6 travel for free.
If you just stay in one region of Japan then you can get much cheaper passes - see the link above for details (eg 5 day pass for £160)
Rail fares in Japan are not cheap. For example, a one way ticket from Tokyo to Kyoto will cost you around £70 each. Kyoto to Hiroshima will also cost about the same amount and then Hiroshima to Nagasaki will cost around £75. Going direct from Tokyo to Nagasaki is around £150 one way.
You can get a regional 7 day pass for £122 that covers the whole area from Kyoto down to Fukuoka and then for another £6 you can get free entry into different tourist places. See here for one agency that offers them (details are in the link below - Seat61)
https://www.klook.com/en-GB/activity/2729-7-day-jr-sanyo-san-in-area-pass-jr-pass/
Going in the other direction, Tokyo to Fukushima is around £50 or Tokyo to Sendai is £55 or all the way to the top, Tokyo to Aomori about £90 one way.
It's these sorts of journeys that make the regional passes (eg 5 days for £160) perhaps more worthwhile than the national Japan Rail Pass.
If you do decide to travel by train there's an excellent site that covers how Japanese railways work and how to find timetables etc here (he also covers lots of other countries as well:
https://www.seat61.com/Japan.htm