i want to know that if i use atmega328-pu instead of atmega8 on this design provided by "https://www.arduino.cc/en/Main/ArduinoBoardSerialSingleSided3", will it works fine on this design or i have to use only atmega8 ?
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.