I know and I can program any AVR microcontroller with Arduino IDE and Arduino Uno board. Can I use Arduino board with AVR Studio? Or I have to buy an AVR Development board?
You can use Atmel Studio to write code for the MCU(s) on the Arduino boards. If the IDE itself does not support uploading the code to the Arduino then it is still possible to use AVRDUDE or BOSSA to upload it via the command line.
There is an Arduino plugin available for AVR Studio, which allows you to upload sketches to an Arduino, just like you would with the Arduino IDE. I think this also enables you to upload plain C/C++ files created with the standard AVR tools.
Alternatively, if you have an AVR ISP programmer, you can use this to upload compiled C/C++ to the Arduino from AVR studio without needing the Arduino plugin.