For such a short visit to Kyoto, you will need to plan your transportation/walking routes to make the best use of your time. Get a guidebook with maps and plan accordingly. Also, be sure to stop at the Tourist Information Center, either at the airport or on the 9th floor of the Kyoto Station building, for specific tourist maps and information. The TIC at Kyoto Station will also be able to advise you of any local closings or other problems. Kyoto Station, particularly from the top floor, is worth a photograph or two in itself.

In no particular order, but concentrating mostly on the east side (like Pickles, I think that is where you get the most bang-for-the-buck), I'd suggest:

Sanjusangendo Hall - just east of the station area and on the way to Higashiyama.

Kiyumizudera - the one "must do" but will be crowded even in December.

"Pottery Slope" and Sannenzaka - just north of Kiyumizudera.

Nazenji - a short walk further north.

Heian Jingu Shrine - should be pretty in winter. If there is snow (unlikely) it will be beautiful.

Ginkakuji (Silver Pavillion) - at the north end of the Philosopher's Path. If you walk the whole thing, this is where you come out. Might be easier to take a cab or bus.

Nijo Castle (Nijo-jo) - take a cab or subway to get across town. This is far better that the overrated Imperial Palace.

That is more than you can comfortably do in one day but if there is extra time, consider a cab to Kinkakuji (Golden Pavillion.) Ryoan-ji , with the famous Zen rock garden, is a short walk away.

Have a great visit.