Computer monitor shuts off temporarily when the window AC kicks on

Solution 1:

The voltage is dipping when your air conditioner kicks in. This will happen if your PC and/or AC is a long distance from your breaker panel AND they are on the same circuit. On start up, the AC is drawing more current which means the voltage drop to the wire or I2R losses goes up decreasing voltage at load and hence the monitor shuts down.

From Fundamentals of Electricity - Voltage Drop - Definition

Without knowing the info on your UPS, but I'd guess the voltage never goes away so your UPS does not kick in.

The true solution is thicker wires and/or separate wires to the air conditioner and the computer. That is not realistic for most users.

Alternatively, the air conditioner and the computer must be on different circuits. Move your computer to different outlets in your home and see if this impacts the problem. All outlets in the same room are typically on the same circuit. But if another outlet is closer to the breaker box, the impact of AC starting will be less. Extension cords will make the problem worse.

You do need to investigate the problem a bit more. This info should help.

If moving the computer has an impact on the problem, then the problem is voltage dip and you can select a solution to fix the problem. Options for solutions are outlined in: Case Study: Choosing among Voltage-dip-compensation Devices on the Market.