74LS47 is an Open collector driver, you will connect the ABCDEFG outputs to the individual cathodes of the display. Use current limit resistors:
(5V - (Vce of PNP) - (Vf of LED) - (Vol of 'LS47)/current = resistor, example:
(5V - 0.3V - 2.2V - 0.27V)/.02A = 111.5 ohm, use a 120.
Plug in the numbers for your selected parts.
You will need 4 PNP transistors to source current to the 4 common anodes of the display.
Connect 4 Arduino outputs with 220 resistor to the 4 PNP bases. Collector to +5. Emitter to the common anode.
Connect 4 arduino outputs to the ABCD selection pins on the '47.
Then write multiplexing software using blink without delay programming:
Output the '47 digit #1 info, turn on PNP #1 by bringing its base low. After 2mS, turn off PNP.
Output the '47 digit #2 info, turn on PNP #2 by bringing its base low. After 2mS, turn off PNP.
Output the '47 digit #3 info, turn on PNP #3 by bringing its base low. After 2mS, turn off PNP.
Output the '47 digit #4 info, turn on PNP #4 by bringing its base low. After 2mS, turn off PNP.
Repeat.
During the 2mS, do stuff like read the coins, check serial port, etc.