I'm wondering is it possible that a monitor will become slower in its warm up speed simply due to age.
Or, like the computers memory, is there a way to speed up the warm up time for the monitor as well?
You have a tube type monitor. The warm up is taking longer or the emitting surface is getting worn out or both. Slight chance that whatever source feed the filament is out of spec. or that something is wrong with the high voltage circuit.
That slow warmup usually indicates that the tube is near the end of its life.
In the old tube type TV days we would have have hung a booster on the tube that basically stepped up the voltage to make it work better. That shortened the life but since the tube was worn out anyway, what the heck.
In Your case Budget / be prepared for a replacement monitor in the future.
I highly recommend you do not open the monitor. The high voltage can linger on and knock you across the room or worse.
Sorry to give you bad news.