I have a small (13"?) TV with only standard video and mono-audio inputs. Will a Playstation 3 work with this?
It does work, but you may have trouble navigating the game. I use my PS3 on my 20 inch CRT (I believe this is the same type you are talking about) and many games are almost unplayable. An example would be Way of the Samurai 3, I cannot read any text, menu or otherwise. It isn't a functional problem, it is just the way most games are made these days.
It will work but as Dark_arc said, you'll have trouble reading text since the TV has to cramp the high resolution images into a small one it can display,
I used to play on a 17" CRT.