Yes, that the vacuum operated actuator that switches a flap inside the air filter housing tunnel. If you connect a rubber pipe to that nipple and suck on it quite firmly, you should be able operate the flap. That would tell you immediately if the diaphragms in the actuators are good or not.
IF you say the car takes very long to warm up, it seems to me that the thermostats were removed!! This is bad as this also means that the bypass circuit is not closed off. It is therefore high priority to get those thermostat covers off to see whats going on there. Correct thermostat: C28067#
Other than cooling system (and brakes), the ignition system is the most important aspect to look at. Again, judging by the condition of your car, I would not be surprised if its still fitted with the infamous "OPUS" ignition system. If so, that MUST come out immediately and be upgraded with something modern, either the ignition system from an XJS (cheap if you can find a system - dissy, AB14 ignition module and dual coils) or from Lumenition. Post a pic of the valley of the engine from dissy to waterpump please. Also send a pic of the flying saucer thingy on the side of the dissy.