Hook a video signal up to it then shine a flashlight at it sideways to see if you can make out anything on the screen, it could be the monitor is working but not the backlighting (tubes or inverter circuit).
Don't know what to make of the brownish capacitor, we'd need a good picture. Could just be they used some cement/glue/etc on the board and heat turned it brown over time. "Usually" if a cap problem is bad enough to make the monitor display nothing from unstable logic circuits, the PSU protection circuit shut it down and the monitor LED flashes or changes from "on" to standby color, if not shutting off completely.
With a multimeter you could measure output voltages from the power supply board. Be careful, considering the inverter circuit on CFL lit monitors produces a few thousand volts. If shining a flashlight doesn't let you see any image, try unplugging the CFL tubes and try again, maybe the inverter has a short and without a load it stays working, but otherwise the main logic board may be damaged.