Do you plan to stay a full seven days? Where do you plan to stay? Where do you plan to go?
Depending on those answers, the amount of cash required could vary quite a bit. I'd even argue that a JR rail pass may not even be necessary. On our last couple of trips, we found that having a rechargeable Suica card to be a better use of our money. For example, we took the Odakyu rail to Kamakura and the Tobu rail to Nikko. There were also plenty of smaller lines within Tokyo that are not covered by the JR rail pass. If you plan to ride the shinkansen, definitely get the JR Pass. We first bought our Suica card in 2008 and have saved the remaining balance for return trips. The Suica card stays good for a 10-year period and that period resets the next time the card is used. The Suica card can even be used for drink/food purchases around the train stations. The other big IC card is Pasmo which used to be centered around the Kansai region but is now interchangeable in the Suica regions.
As for food, most conbini (convenience stores) take credit cards. We found that our local credit union offered a MasterCard which had no foreign transaction fees and no currency conversion fees. The topper was that we accrued bonus points for travel expenses. So, if you are at all familiar with the Japanese conbini, there will be no shortage of opportunities for some kind of meal via credit card.
That said, there are still many places that take cash only including many of the small restaurants that surround the train stations. If you don't mind ramen, tonkatsu, curry, soba, etc., many meals can still be purchased for between ~$5 and ~10 depending on your tastes. Some of these restaurants use vending machines to dispense the meal ticket for your order so that you do not have to even talk to the staff. I think most street food, i.e. the kind you find outside shrines during festivals, will cost ~$2-$5. To be more exact, onigiri in the conbini were ~$2 and chips like Jagabee cost about the same.
If I were pressed, I'd say that I could feed myself for between $10 and $20 a day. There's always instant ramen right? For better meals, I'd say you'd be looking to pay between $30 and $50 a person. I spent about $30 for sushi one time in which I ate a lot. For our yakiniku meals, I believe I spent about $50. Though, don't be fooled! The cow tongue was a very cheap option and tasted great!