How to interface with micro LCD projector?
I count 11 flat conductor connections to the LCD.  That is just enough for a 7 x 4 display (one of the 12 wires is likely surplus, unconnected or common with another).

The first digit only has to show three states 0, 1, 2 (and off) and this can be arranged with list 4 driven elements that have a couple of segments connected together.  This leaves 3 spare elements that can be driven for the colon and the AM and PM annunciators.

As a raw multiplexed LCD display it will require multilevel drive and this can be done with most microcontrollers at the expense of extra pins and resistors.  If the AC drive logic is not done in hardware then it has been proposed that blocking capacitors are used on the row or column drives so that there is no residual DC to cause display degeneration.

Testing with DC is usually safe but generating AC from a PC Earphone output is also usually enough to drive simple displays.  Use a tone generator at a low frequency of 100 to 400 Hz and text the various pin combinations briefly.

Often having two wires of 30 to 100cm laid left and right on your desk are enough to test a LCD between them because of stray capacitive mains pick-up as the LCD has such a high impedance.

EDIT:
Your display is a 1:4 multiplex as there are not enough connections for a simpler 1:2 or 1:3 multiplex. Direct drive (1:1 or non-multiplexed) is the simplest but requires one pin for each individual segment (28+1 connections) because it has a single common backplane.

Here is some reading material.


https://www.circuitvalley.com/2012/02/pic16f917pic16f1907-7segmentlcddriverex.html Discussion of drive voltages
What is this type of digits-only LCD called? Processor options, some PICs also have built in drivers
https://www.nxp.com/docs/en/data-sheet/PCF1178C.pdf 2:1 multiplex
information
https://www.intersil.com/content/dam/intersil/documents/icm7/icm7211am.pdf
older 4 digit driver IC