1) wheather we can enable a serial interrupt when the ADC interrupt is in process.

You can do that. However, if you need to use interrupt nesting, as this is called, you should ask yourself why your interrupt service routines are so long that you need it and what you can do to make them shorter.

In general, interrupt routines should be short enough that nesting is not necessary unless you have very, very strict timing requirements (which serial communications doesn't).

2) How can we enable the priority level of ADuC 848.

The prioritiy of each interrupt can be selected by setting/clearing the appropriate bit in the interrupt priority (IP) register.