How to effectively make a self-sustainable arduino project

You need a diode between the solar panel and battery to make sure the battery doesn't discharge into the panel if/when the sun goes down (currently under dispute). Then you also need a shunt regulator to make sure you don't overcharge the battery; I'd suggest using a TL431 which is a pretty neat/simple chip for precision voltage control. An alternative to a TL431 shunt regulator would be to monitor the battery voltage with your Arduino and program the Arduino to burn current (light up LEDs, whatever) if the battery is overcharging.