This video shows how to replace a SSD in the XPS13. It looks like either a custom made SSD or at least a special type that has no casing.
Is it possible get a part number and buy a bigger one?
That's a standard mSATA SSD. You can get a 256gb Crucial for just over 2 bills at Amazon. Mushkin just released a 480gb mSATA drive, but that one's going to cost you a bit more.