How can I ensure that I have a mode C transponder?
For any aircraft you should check the accompanying POH for a full equipment list of what the plane has been outfitted with. Likewise you should be alert for any place cards noting defective or inoperative equipment. You should also check the aircrafts logs to make sure the transponder has been properly inspected according to § 91.413 ATC transponder tests and inspections and that it is operable. Just because you have one does not mean its legal to fly with. A transponder capable of Mode-C (altitude encoding) needs to either be connected to an altitude encoding altimeter or a blind encoder to report altitude. You should be able to find this out in the POH, if its not connected to one of these two units it will behave like a Mode-A transponder. You can find a  discussion on the various setups here. If you are really curious about how Mode-A/Mode-C communication works this is a pretty good read. 

Specific to the Garmin unit you have pictured you can get it to display the aircrafts altitude by pressing the "func" button (pressure altitude that is) and compare with your altimeter. If the unit has a readout it then an pressure unit can be assumed to be connected. NOTE: This is not an official test by any means.