This would be a good place to start for getting the audio data to Linux: http://www.malinov.com/Home/sergeys-projects/compact-usb-audio-adapter. You'll have to use a Linux image with the audio driver, but Sergey also has an excellent blog post about it along with the patches for download: http://www.malinov.com/Home/sergey-s-blog/intelgalileo-buildinglinuximage. This should be the extent of low-level Linux programming you would need. Once the data is sampled and stored in a file or whatnot, you can manipulate it with C/Python and shuttle it to the Arduino sketch if needed.
The AD7298 might be spec'd to 1MSPS, but since the default Galileo OS isn't real-time and the board doesn't have a buffer for ADC samples it'll never come close to 1MSPS.