Temples will be fine across Kyoto and the region including Nara as New Year is an important time for the Japanese to visit Buddhist and/or Shinto temples. Museums though are likely to be closed I suspect from the 29th as I suspect will the up market restaurants outside of the top hotels. If you focus on those up to say the 29th and save the temples etc for after you should be fine. Temples on the 31st night and 1st are busy but really fun with some having lots of regional snacks for sale in the surrounding area.
I do suggest you book tables in your hotel on the 31st for evening meal as thats the time your likely to struggles the most to find somewhere to eat.
We are in Kyoto during some of that period but have an apartment hotel like we did last year as this gives us some flex in terms of eating in or out.