Chosen Solution
I have an old NEC CRT Multisync C400 Monitor that has been sitting in an outdoor shed and is now a bit rusty and dirty. However everything looks connected and no capacitors are bulging. When I plug it in and press power nothing happens at all, and I have already checked the fuse.
This is not an answer just a precaution. Please note that these old CRT monitors have very lethal, high voltages inside and so one must be very cautious and careful! If you don’t understand this stop.
Rust = IRON oxide = conductor If you manage to light it up, there is no telling what the affect of that iron will be. But if you want to try, it is somewhat obvious where that AC input is and goes. It is a switching PSU. It should be easy to OHM out the circuit thru the switch on the left to the bridge rectifier which may be that black component below the transformer on the bottom right. Past that, use your diode function to check any diodes or transistors. If you power it on, stay clear of the device in the top right. That is the HV transformer (flyback). But then, 120V can be lethal also. Dealing with a switching PSU is another story. The best that you can hope for is there is DC off the bridge into it and DC out of it around that large power resistor. If the PSU has a problem, I suspect it would be from the rust. That does not look like something you can sufficiently clean up. My bet, it is toast.
