Modding an old Sony Ericsson W205a phone to work on car
I recommend leaving the phone on a cig lighter charger in the car if your cigarette lighter is powered all the time. The phone should use very little current when the phone battery is fully charged. You can check this by measuring the current the cigarette lighter takes. It should be less than 20 milliamps from the 12V supply. It would not drain your battery under normal conditions (driving once a week at least).   

Here is why I not recommend modifying the phone: Opening the cellphone and wiring something to the On/Off button can cause more trouble than good, the phones are densely packed and you'd also need to design and build a custom piece of electronics to operate the button contacts. Probably not the best investment of time. 

Either way, the calls made to the phone will goto the voicemail when you are not in the car - either because it's off or no one pick it up.