How to turn on LED with only a cathode pin
Read carefully the datasheet. The info you are looking for is there.





Therefore the two decimal point LEDs are connected between pin 1 (LED supply, common for both) and the two cathode pins (4 and 10).

Then it's up to you how to power them up. If you want to control them independently, you just wire pin 1 to a suitable positive power rail and put any switching device between each cathode and ground. It may be a BJT, a MOSFET or even a relay. If you want to control them via a MCU GPIO pin, you can also connect it to that pin(s), as long as the MCU can sink enough current (of course you must add a limiting resistor in series).

Since the datasheet reports a 5mA recommended operating current for DPs:



you may be confident that many modern MCUs can sink that much current (for example a GPIO pin of the ubiquitous Arduino boards, from its ATmega328P MCU, can handle it without problem).

For example:



drive the GPIO pin LOW to light up the decimal point whose cathode is connected to pin 4 of the display.