Thanks guys! I see that arduino UNO's 3.3V supply has 50mA maximal current rating. So If I use a 75 ohm resistor to limit the current then it takes about 50 seconds to charge up to 95% (3.135V). Then I will leave it connected to the capacitor as long as the arduino power is on. I'm not sure about feeding current back to the supply when the arduino is powered off. Should I add a diode? But then I take away 0.6V from the supply voltage and can only charge up to 2.7V. Then the capacitor will also last half as long (3 days, still long enough for me).
CrossRoads, Is there some diodes with less turn on voltage for this type of work?
RuggedCircuits, is the self-discharging due to air?