Measuring SuperCap current leakage avoiding voltages drops in ADC
One important thing you didn't mention is whether the voltage you are measuring is already within the A/D range, or if it needs to be attenuated before being presented to the A/D.

If no attenuation is needed, then just connect the cap directly to the A/D input.  Make sure there isn't anything else connected to that input.  The leakage current of the A/D input is then your discharge current error.  You have to look it up, or course, and compare that to the expected discharge current.  Most likely, the A/D leakage current is much smaller than what you are trying to measure.

If attenuation is needed, then you could buffer the cap signal with a opamp that has high input impedance, then attenuate the output of that.  Or, you can use a P channel FET with sufficiently low off-state leakage to connect the cap to the attenuator.  This would only need to be done for a few &micro;s every sample, which could be every minute or so.  The average current thru the attenuator is then so small as to be irrelevant.