I'm not sure how to interpret your question.
Yes, you can build a digital clock using an Arduino. It would (or could) share a lot of principles with the article your reference. Essentially, you could replace each chip with a software subroutine:
void tick()
/* Execute this function every time we see a one-second clock tick */
seconds = seconds + 1; // count seconds like a 7490.
if (seconds >= 10) { // counter reached limit?
seconds = 0; // wrap
tenseconds = tenseconds + 1; // count 10s of seconds like a 7490
if (tenseconds > 6) { // divide by 6
tenseconds = 0; // wrap
minutes = minutes + 1 ; // count minutes like a 7490
// etc
}
}
// We have the time in the appropriate digit variables.
// convert to displayable form for 7-segment displays
digit0 = digit2segments[seconds];
digit1 = digit2segments[tenseconds];
digit3 = digit2segments[minutes];
// etc.
updateDisplays(); // actually send out signals t the displays
}
No, you can't replace a handful of 7490s and 7447s with an Arduino directly; to start with there just aren't enough pins.