18V/5W solar panel to trickle charge a 12V car battery?
With a conventional solar setup, solar panels charge deep-cycle batteries. A charge controller is used to prevent overcharging and correct voltage and current. Car batteries are for a surge of current to start the vehicle. I'm not sure if a conventional charge controller could charge your battery correctly because deep cycle and car batteries have different chemistry. But from what I found Here, you should be able to. 

I would just try it with a charge controller, or try to make one. Just experiment with some different setups and find something that works for you.
Also, I don't know if 5w could power your battery because it depends on the capacity and the current draw from the Arduino.