do i have to use delay to do this?
No. What that picture is telling you is that you need to notice when the pin goes LOW, and record when that happens. Then, you need to notice when the pin goes HIGH again, and record when that happens. When the pin has gone LOW and then HIGH, you use the recorded times to determine how long the pin was LOW. The interval tells you what kind of coin passed the detector.
It is unlikely that you'll be able to notice when the pin changes state without using an external interrupt pin and an interrupt handler for the FALLING edge and another one, on another pin, for the RISING edge, or one handler on one pin for CHANGE. (In that handler, you'd need to determine if the state of the pin is now LOW (it was a FALLING edge) or HIGH (it was a rising edge).