Write a program to read the eeprom and write it to the serial port. EEPROM doesn't get mondified when you download new programs. There's only 512 bytes of EEPROM, so just copy and paste it into a text file.
If the contents of the EEPROM are binary it'll take a bit more work.
If you have a programmer I'm pretty sure it can be used to download the EEPROM.
Tell us a bit more about what you're trying to accomplish and maybe the answers can be more helpful.
-j