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.