Some devices that output composite video have capacitively-coupled outputs, but many have resistively-couple outputs; nearly all composite-video inputs are capacitively coupled. If your device has a resistively-coupled output, you're in luck; all you'll have to do to detect sync is to use a comparator to detect whether the signal is above or below a certain DC threshold (sync will probably be at ground potential, and the lowest non-sync signal will be well above 0.1 volts, so 0.1 volts may be a good threshold).
Note that in addition to detecting sync, it may be necessary to also perform DC level restoration; some monitors may perform it automatically. On a resistively-coupled output, black will probably be represented by a DC level of 0.3 volts. Some monitors may expect a black signal to have a DC level of 0.0 volts, or they may expect to sample the signal level shortly after each sync pulse and regard that as black. Depending upon the design of the monitor, you may get acceptable results feeding in your video signal as the signal level, or you may find that black becomes an annoyingly light gray.
One final note: composite video typically uses 75 ohm DC termination. Many devices which output composite video generate a signal whose voltage is twice normal, but then feed it through a 75 ohm resistor, and many devices which input composite video have a 75 ohm resistor to ground. The analog inputs to your monitor may or may not include such termination resistors internally; connecting them may thus affect the signal level.