Supercaps self-discharge in hours/days, no use for a timepiece. They are best for very medium/high
currents for short times, totally opposite of lithium batteries! The varying voltage requires a suitable
SMPS to manage one too.
Back to your maths, if the displays are taking 0.5mA per segment, you need to look at what the
rest of the circuitry is consuming, the display might not be the dominant factor.
4 LEDs for 3 seconds is the same as 12 LED's for 1 second.
12 LEDs * 500uA = 6000uA/s
6000uA/s * 50 times a day = 300000uA/s = ~83 uA/h
You don't mean "uA/h" you mean "uAh" - no division anywhere.
You are over-complicating your approach here - charge = current x time, simple:
4 LEDs = 2mA, 3 seconds 50 times a day = 150 s, therefore charge is 0.002 x 150 = 0.3C
1 mAh = 3.6C (because 1Ah = 1A for 3600s), so 0.3C = 0.083mAh per day (so you got the right answer).
What you need to also measure is the current when the watch is off (presumably there is a RTC of some
sort (these take about 0.5uA on battery-backup or so, which is 12uAh per day - that is typically running
from a separate button cell though.
Are you going to see anything with just 0.5mA going to an LED?
You will need high-brightness LEDs at that current level, and you shouldn't expect it to work
Good polycrystalline silicon solar cell yields about 0.5V and 30mA / cm^2 in full sun. Derate by a factor of
10 for overcast skies, 4 for winter, so expect 0.8mA/cm^2 averaged at 0.5V, and you'll need about 6 to 8
in series to get the voltage, so 100uA/cm^2 at 3 to 4V... (But check my arithmetic yourself!)