Will Apple TV 4K work on my 2010 Sony XBR-52HX909?

Yes. I expect that the ATV4K will plug into one of the Sony's HDMI ports, 'shake hands', and determine the best 1080p signal the old TV can handle. In the ATV4K's Settings app's Video and Audio section, you may be able to tweak the defaults for a superior chroma value and calibrate the color with the help of a modern iPhone - I'm not sure (never messed with a 1080p TV and the ATV4K).

Likewise if you've got an AVR or Soundbar in between.