I've banged on about this before, but unless you have very specific reasons to visit Tokyo, I believe, certainly with your budget, that you'll get an awful lot more out of your trip ignoring Tokyo entirely. And I say this as a confirmed Tokyophile - I adore the city.

Cherry blossom time is incredibly specific to each area and changes from year to year. Best get it out of your head now, there is 0 chance of sakura. What you might get is some Autumn foliage in November and the chance of snow if you're willing to go into the mountains, particularly in December.

Did you mean this November/December or next year?

Let's break it down
1 person - 10 days - $1,000 (roughly 100,000yen)

Accommodation - 2,500yen a day upwards (25,000yen) MINIMUM
Food - 1,500yen a day upwards (15,000yen) MINIMUM
Admission and entertainment charges - 1,000yen a day upwards (10,000yen)
Travel on transport not covered by the JR pass (unknown - but let's say 5,000yen)
JR pass for 7 days - 28,300yen

Total so far: 84,000yen ($840) I stress that this is rock bottom.

And I can promise you that you will find ways of spending that extra 1,600yen a day

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So, this gives you the most very basic of holidays.

You can step up your experience, get better hotels, spend a bit more on food, have more for snacks and trinkets, all within your budget.

To do this don't buy the JR pass and take Tokyo and Hiroshima out of the equation, concentrate instead on visiting the areas around Osaka and Kyoto. I assure you there is plenty to do for ten days in and around the Kansai region with short trips to Himeji Castle, Mount Koya, Nara, Kurashiki, Naoshima (Island), and, if they still exist then, a stay at one of the Okayama Villas such as Hattoji. All of these will provide absorbing, affordable and memorable excursions.

KIX would be the best airport, but even NGO lets you travel to and from Kyoto/Osaka cheaply. To make the most out of your budget, avoid NRT.

Come back to Japan when you have more time and spend at least a week in Tokyo then.