In my project I am using Solar panel(12V 5W)to power up an Arduino Mega. The DC battery which I have is of 12volts. Can I use it to power my Arduino board? If no can I use a simple resistor to decrease the voltage. In such way what should be the value of the resistor so that the voltage may come down to 9 volts or lower.
You can power the Arduino with 12V through the Vin pin or the power jack. Bear in mind that the current that the 5V regulator can supply is limited. The more voltage that the 5V regulator must drop the less current that it can supply and stay under the practical limit of 1W dissipation. A resistor is not good for dropping supply voltage as the voltage that it drops is dependent on the current it must pass. A switching regulator to drop the 12V to 5V would be the best solution, in my opinion.
Or put 2-3 diodes in series with the supply to drop bit of voltage ( .6 volts each )
With only 5W of solar power, efficiency is going to be very important. A resistor, diode or even the Arduino's onboard regulator are going to waste about 60% of the input power. That would be expressed as 40% efficient.
A switching regulator, such as this one from Pololu, will be at least 80% efficient. You will get more than twice as much useful power from your solar panel. Connect it directly to the Arduino's 5V pin, bypassing the onboard regulator.