How can I set up SPI using Atmel Studio and C code for ARM processor?
You have two options here.  You can manually set the configuration registers directly for the SPI peripheral, or you can use the "Atmel Software Framework" which abstracts a lot of the details into a higher-level API.  In either case, you will have to deal with a lot of things that the Arduino environment handles for you.  

You'll need to setup the processor's clock, which means selecting a source (Typically you have the option of an internal high speed, internal low speed, external high frequency crystal, external low frequency crystal, or external clock) and a combination of multipliers/dividers that derive your desired core clock frequency.  Then you need to ensure that the SPI peripheral and the GPIO modules receive a clock signal (by default, clocks are not routed to any peripherals to save power) and that the clock signal is divided down to an appropriate speed (the MCU data sheet will specify min/max timing, as will the data sheet of your external IC, and you'll need to make sure you satisfy the requirements of both).

You'll need to properly configure the GPIO pins to enable the SPI peripheral to control them.  Many MCUs of this class will have several SPI units, and the datasheet will specify which pins can be used by which unit.  Most pins will have many peripherals which can be selected, so you'll need to set the pin multiplexer to select the proper SPI unit.  You'll also need to set pin direction and pullups/pulldowns as needed.

Finally, you'll need to configure the SPI peripheral itself.  This will include setting clock speed, which clock edges to use, master/slave, pin routing, and many other options that will be spelled out by the datasheet.

If you're just starting out, then ASF is the way to go.  There are examples in the ASF documentation linked in previous comments that should help you out, and if you examine them closely you should see that the example applications do all of the things I've outlined above.  However, the ASF documentation does an exceptionally poor job of actually explaining what's going on and why, so you'll need to carefully study the MCU datasheet sections on clock, power management, GPIO, and SPI to really figure out what you're doing.  Unfortunately the datasheet also does a poor job of explaining how peripherals are meant to be used, so some reading between the lines will be required.  As a general rule, vendor APIs and documentation for ARM MCUs (not just Atmel's SAMs) are not very good, so a lot of reading and experimentation will be required.  Use the right click->"Go to implementation" feature of Atmel Studio liberally to learn more about what ASF is doing under the hood.  Referring back to the datasheet descriptions of the various hardware registers' functions will help.  

So all of that is a very general answer--your specific solution will depend on the particular hardware setup you have, so you'll need to look at the Arduino Due schematics to figure out which pins you'll need to use for SPI and what clock settings are appropriate.  The MCU will start up running from an internal oscillator by default, so I'd suggest you start by just getting the GPIO working--just blink an LED to make sure you have the hang of the GPIO, then start working with the SPI module.  

Above all, remember that the datasheet is your friend!