How to power up this LCD 7 segments display
that's a controller-less dynamic (mutiplexed) display, with 4 coms.

So you will to program the mcu to do a lot:

1) the com pins are driven with 1/4 dc signal sequentially;

2) the seg pins are driven with 1/3 bias voltage. out-of-phase to be dark and in-phase to be transparent (not lit).

the way to do it is to:


put segment info in a buffer;
write a display update routine that scans digits and updates the com/seg pins according to the current digit's segment information;
call that display update routine periodically, preferrably from a timer isr.


the display update routine would be similar to an led display routine - an example of that (for led) would be here (https://dannyelectronics.wordpress.com/2017/06/30/a-more-flexible-driver-for-7-segment-leds/), but more complex due to the 4 com pins and bias generation. 

it is much easier to pick a mcu with built-in lcd driver and program that mcu as a display controller / slave taking in information from a master (in this case your arduino).