You will need to make sure the code you use is as efficient as possible - the AstroEQ firmware is already pushing the capabilities of an ATMega. But, by all means try it out and see how you get on.
There are a few key parts that you will need to change to get your firmware working.
Line 345 is the following:
if ((decoded == -2) || Serial_available()) { //is there a byte in buffer
You will need to add a flag that indicates when your Ethernet code has a character available.
Line 348 is the following:
recievedChar = Serial_read(); //get the next character in buffer
This is where the next character is read. So your Ethernet code will need to have some function which will return the next character in its buffer.
Line 354 is:
Serial_writeStr(decodedPacket); //send the error packet (recieveCommand() already generated the error packet, so just need to send it)
This line writes a null terminated string to the serial port. The Ethernet code will need some function to do this.
You can replace these three lines with some code that has an equivalent function but which uses Ethernet. Now because there are two sources (Serial and Ethernet) you would need to select which one to use, but my suggestion would be to drop the Serial stuff from your variant and stick to just Ethernet. If you want to use the config utility you would upload the stock AstroEQ firmware (with serial comms), then configure the mount, then upload your custom firmware (with Ethernet comms).
The alternative would be to use something like a raspberry pi setup to relay information between Ethernet and the USB port of AstroEQ. Or you could go for something similar to
this. These would probably be a more reliable option. The final alternative would be to design a separate Ethernet to Serial circuit - could be another Arduino with your Ethernet shield, which would then do all of the heavy Ethernet processing and forward to the data to the AstroEQ board.
Personally I think the alternative options would probably be more useful in the long run. The ATMega's as useful as they can be aren't really designed for the raw processing power needed to run the Ethernet interface at the same time as doing the AstroEQ stuff - they really aren't that great for multitasking.