> Also, can I just change the ADC initialization settings in my code to access this external one?
No.
Except for the name (ADC), both are very different devices ...
> I don't even know where to begin to write code to analyze this external ADC. Is there an example project that I can use?
The AD798x is a SPI device, so you would want to start from a SPI example.
Looking at the datasheet, there is no configuration or setup to do, you just need to read the results.
Start by selecting the operation mode you like (starting at page 17). Make sure your SPI settings like data size, clock polarity, bit order etc. match those in the datasheet, SPI is not tightly defined like e.g. CAN.