As I don’t know you, the best advice I can give is a summary of what I would do in your place.
Get the Keisei line to Aoto and then continue to Oshiage (9 mins). (approx 1hour 15 mins total – the 06:45 & 07:30 Keisei skyliners should actually stop at Aoto, according to their holiday schedule, but this isn’t shown on www.hyperdia.com) then it’s just one stop to Asakusa. I guess you’d arrive by 8:30am. Roam around Senso-ji for a while to see the temples and be there when the shops open – make sure you grab some freshly baked rice crackers! (I can’t imagine that nearby Kappa bashi AKA ‘kitchen town’ would be open today, but it might be worth asking). From about 09:45 the Sumida river line runs boats down to Hionode Pier (40 minutes, approximately every 40 minutes http://www.suijobus.co.jp/english/cr..._su_line.html), stopping off at Hama Rikyu just before hand – a beautiful Japanese landscaped garden (660yen + I think, an extra 100 yen for the garden - if it actually open today).
From Hionode pier you can catch another ferry across the harbour to Odaiba (http://www.suijobus.co.jp/english/cr...ex_f_line.html) which is packed with brand new buildings and developments – very futuristic but with a vaguey 50s bent… apparently. It’s supposed to be VERY popular with young Tokyo-ites – but I haven’t been here yet – one of the reasons that I would choose this itinerary for myself!
One thing I most certainly would like to do on an 11 hour layover is visit Odaiba’s brand new ‘Theme Spa’ Oedo Monogatari (info here: http://club.nokia.co.jp/tokyoq/weekl...ole/onsen.html if you’re a bloke – don’t forget to bring some swimming trunks!). It’s 2,700yen and is supposed to be open every day and, unlike most public sentos which don’t open until about 4pm, this place opens at 11am. Having bathed in a kitschy faked up reproduction of Old Tokyo and taken in the sights of Odaiba, I would cross the rainbow bridge on foot (at least 40 minutes - but you can get a train back) and then make my way back to NRT from Tokyo station or from Ueno – having stopped for lunch/dinner and gotten a decent bento for the plane.

And look out for mystery lucky bags – they don’t officially go on sale until the 1st, but I’ve been sold one before when I explained that I was leaving the country that evening. If you’re fortunate you’ll get loads of presents all in one ‘hit’.

VERY IMPORTANT!!! The Japanese Post offices and their ATMs close down for a few days at New Year, and ATMS at places such as Citibank are wildly unpredictable. I would strongly urge you to change your money before you arrive, especially as you'll be arriving so early. (10,000 should just cover everything - 20,000 will give you some spending money). Please don't get caught out - your credit card may well prove useless!