Sorry, I don't know but you need to pay for admission to the villa. Where I would get the info would be on the second level Information Office in the Kyoto Station (North side), these people bailed me out more than once in my ventures. Further that would be where you would catch the local train going out to Arashiyama anyway. Christmas and New Years are both holidays there with the latter having more importance, my guess is NO it won't be open; but i've been surprised in the past.

There is quite a bit of walking in the garden. Spend about 45 minutes to an hour there so that you can enjoy the views from the top of the garden and the complimentary matcha.

It’s really not hilly at all. There are some inclines but nothing I would call a hill

I took my time enjoying the peacefulness and the beauty of the gardens. I took a lot of photos of the flowers and greenery... spent some minutes too at the photo exhibit (I don't know though if this runs all year). I would say 1 1/2 hours of leisure strolling should do it. It is worth the visit.

It's a pleasant 15-min walk back to the station. Follow the bamboo-grove-lined sidewalk back to The main street in Arashiyama (the one lined with shops and restaurants). From there it's a short walk through a residential area to the station. I recommend printing out the results from Google Maps. You can easily pull up the directions online, but Saga-Arashiyama station also has free maps in English on a shelf as you exit the platform.