Congratulations for ryokan-hunting!
Amazake is a traditional sweet, low-alcoholic drink and drank mostly in New Year's Day.
If you want to see the most well-known event, go to Yasaka shrine in the evening. People come to get a sacred "flame" to cook Ozoni (the first meal of a new year). It is said that eating Ozoni, cooked with fire from Yasaka shrine repels evil spirit and makes you healthy in the new year. After watching this, you've better leave the shrine to hit a temple for bell ringing. Yasaka shrine will be paaaaacked with worshippers!
BTW, this event is called Okera Mairi.
As for bell-ringing, the most famous one is Chionin temple. Its bell-ringing ceremony is always aired nationwide but visitors are not allowed to ring a bell. Kiyomizu temple and Nanzen temple also open for bell-ringing but I don't know if you can get a ticket (for bell-ringin) as they start issuing a ticket in advance. Try small/medium temples and you'll have a good chance to ring a bell yourself. For example, Sinnyodo, Eikando, Honenin, Chisekiin...
Though I've never gone there on 31st, I personally like Kodaiji-temple for its coziness and beautiful garden. It also let visitors to ring a bell.
When the sun rises, worshippers rush to shrines. Yasaka shrine, Heian shrine, Fusimi Inari shrine are really packed. If you want to quietly pay a respect, again visit s/m shrines. Shimogamo shrine near Demachi Yanagi station is very nice with trees and the Kamo river.