Replacing atmega8 with atmega328 on Arduino Single-Sided Serial Board
Yes and no. It will work on this board, but won't get programmed by arduino IDE directly.

ATmega328 and ATmega8 are fully pin compatible. ATmega328 is like bigger and powerful in terms of onchip resources. However, if you want to program ATmega328 on the arduino board, you must install bootloader. Otherwise you will have to program it everytime using ISP programmers such as USBASP. This will connect to the ICSP socket shown on the above board.

Refer to: http://tutorial.cytron.com.my/2011/09/30/how-to-program-arduino-by-using-avr-usbasp-programmer/