I stayed at Sakura Guest House with a friend during Hogmanay. It was only a 15 minute walk south of the Royal Mile and offered great hotel-style amenities, including towels, unlimited tea and coffee, and a private ensuite. Close to hip bars and dine in/take away spots. Ayas and the staff were very polite and didn't intrude. I would definitely stay here again if I'm ever back in Edinburgh for New Years considering it was much better value than central hostels during this period.