Would this be a good solution?
Absolutely not: that circuit is doomed. Sooner or later the car will
generate a voltage spike that fries it.
But there's a simple protection circuit that will almost completely eliminate that risk.
Start with a diode (any old 1N4000-series will do) in serieds with the positive supply to protect against negative spikes. Follow that with a series resistor chosen to drop about 1-2V at the maximum expected current draw (i.e., if you're going to use about 200mA, use an 8-10 Ohm). Then add a Zener in the 20-30V range to ground. The resistor and Zener will absorb any big positive spikes.
If you replace the 7809 in your example circuit with a 7808, you'll have a nice clean 8V going into the Arduino while the car is stopped or the engine is running. However, while the engine is being cranked to start it, the car's voltage will sag quite a bit (8V is considered "acceptable" under those conditions), so the 7808 will drop out of regulation, and the Arduino will almost certainly reset itself.
If you're not hooking up much to the Arduino, and it's not using much power, you can eliminate the 7808. If you are, it's probably better to keep it, so you're dissipating more of the waste heat from linear regulation off the board.