Sure it's doable, if you can do it
Best place for fans I found is bestbyte.net many fans list dBa (noise levels) Watts, amps, RPM, CFM, etc...
They have cheap good fans, expensive fancy fans, Leds fans, etc... in lots of sizes.
I buy all my fans there and most cold cathode lights, been awhile since I bought anything though, I stock up extra's to save on shipping and have them when I need them not 2 days later.
You may need a charge controller of some type since sun will vary and you don't want to drain the battery too low or over charge it too much, doing either will shorten the batteries life.
You could get fancy and put a light senor on the LEDS to only turn on when room is dark, wire a sensor for fan outside near panels to only turn on fan durring day.
Turning on the fan at night for just a few hours without a switch might be trickey. I think you can get a 12V pull chain switch many places, using one you can just use a chain or string any length and turn fan on/off as needed when suns not out, and ceiling height is no problem then and lots of wireing not needed. Just wire switch on an extra wire parralel to the light sensor circuit to bypass the sensor at night when you want to.
How much sun you get for charging and how long the fan runs will make the differnce on how many watts of solar panel you need.
A 92mm fan I took a quick look at for $1.30, DF129225SH 12v 0.37amp 4.44watts 2900rpm 51.78CFM 29.24dBa
A 120mm fan for $7 may be a better value though, since it's a cheap part of your system that should last for years.
LED 120mm fan, 12V 0.38amps 4.56watts 2,000RPM 78CFM 28dBa
More money for larger fan up front, but for only 0.01amps/0.12watts more power used you move 27CFM more air with slightly less noise. Also this particular fan has built in LEDS in your choice of colors.
So check fans out and see what you need and the best value. Then figure your hours to run fan, solar hrs available, and see what you need for a panel in watts, adjust as needed.
You could also get a fancy variable speed temp controlled fan, the warmer the room the faster it runs, the cooler the room the slower it runs.