An arduino UNO draws 50mA at 5V so it will likely draw about 50mA at 12V. Will that drain your battery while you're not driving? Not likely. You can do some measurements with a multimeter since you have more than an Arduino UNO. Turn off your engine and measure the battery voltage. Then if you can measure again after a day, you can use these two voltages to estimate how quickly the voltage drops. If you are concerned with short circuit, then pick the fuse to match what your circuit regularly draw. Say it draws up to 1A at 12V when everything is running, have something like 2A or so to protect your battery. Your battery can output very large current. Without a fuse if something short circuits, your wires between battery and arduino will melt.