In this part
2 >> 8
2 is supposed to be the name of a 16-bit variable (something that is type 'int').
Then you shift out twice:
shiftOut(dataPin, clockPin, MSBFIRST, minutes); // 1 numeral of minutes goes out
shiftOut(dataPin, clockPin, MSBFIRST, (minutes>>8)); // 2nd numeral of minutes goes out
assuming you had previously converted the minutes into two 8 bit pieces to go out to the individual dots on the minutes dominos.
Can you post the hardware schematic you have hooked up? I think I am getting myself confused,
you are using the time library which stores 2 numbers, hours & minutes, that you are then mapping to the domino dots?