Power microcontroller from super capacitor

From your parameters, your supercap would discharge in 1848 seconds to 1.8v under a constant 12mA draw.

$$Bt(seconds) = C(Vcapmax - Vcapmin) / Imax$$

If it's only active for 100ms every minute it has a duty cycle of:

$$100ms / 60000ms = 0.0016667%$$

It would last ~1.1 million minutes, or about two years. That is excluding the sleep mode draw however. At 20uA, interestingly enough your total active mode power consumption would be about the same as your total sleep mode power consumption, so we can easily estimate that including the sleep mode (which will be 99.84443% of the total time), your device will last for about a year from fully charged to 1.8v. You could extend this quite a bit by adding a high efficiency buck-boost, provided you don't add too many losses with it. Some modern boost converters can yield 1.8v out from as low as 0.25v in.

The answer from Drunken is correct, but there is one important thing missing. You must consider the supercap ESR. For supercaps, they often are in the range of 100 ohms, which would cause a voltage drop of more than 1V when the MCU is active, causing it to shut down.

Therefore, you must have a regular cap with low ESR in parallel, that can hold the voltage during the 100 ms of activity. Something like 1000 uF electrolytic would certainly be appropriate.

Also check the caps leakage. Both the supercap and the parallel electrolytic. This current could be significant, relatively to the standby MCU current. However, they are rarely mentioned in datasheets. You may need to test.