Powering a microcontroller with a small solar cell and a capacitor
Basically, this could work, but it's going to need a lot of components just to have that threshold voltage, then to activate the microcontroller, and then to disconnect it once the voltage is below another, lower, threshold voltage.

The problem is that all these design components will need to be extremely low in power consumption, or things won't work out, and they need to work without an external power source, so coming up with solutions is harder.

However, integrated solutions for your problem do exist: TI, for example, has a supercapacitor solar charger demo board, http://www.ti.com/tool/TIDA-00242.

It features the BQ25570 as central component that takes care of getting the most out of your solar cells (you want to operate solar cells at a specific point in their I/V curve for sensible efficiency), controlling capacitor and system voltage.

Now, be very sure that spending your space and cost and complexity budget on a solar cell is wise – a simple LiIon battery, as used in hearing aids, pocket calculators, or e-Ink store price tags, would probably suffice for more than the life time of your device.