Looking at the schematic you supplied it has a 6 pin ICSP2 socket with each pin going to the correct place so you should be able to plug the 6 pin header of the programmer into it and the programmer should supply power and program your chip.
If your having problems then the LCD also uses pins 11 & 12 so may interfere with programming.
The sparkfun link you posted has links
to how to setup the IDE for programming.
Unless you need a 16MHz processor speed you could reduce the component count further by removing crystal (Y1), capacitors (C1 & C2) & resistor (R2) and setting fuses to use internal 8MHz clock.