Poor Man's UPS for a Microcontroller
Based on the datasheet of the uC that will be used, its maximum current draw is 150 mA. I'll be assuming that this is at 3V3, which will equate to around 33 Ohms. Will this assumption be correct? Can I use this value to determine the Capacitor discharge rate?


No you can't (reliably)


  ..which means that the capacitors will keep my uC alive for around 8 days. Are my calculations correct? Or are there other things that I should be aware of?


No your calculations are not correct.

You are forgetting (or neglecting) the fact that at the same time you are consuming current from the capacitor the voltage across it is decreasing linearly, in other words, once the capacitor is fully discharged then the voltage across it is also zero, meaning that you will run out of useful supply voltage long before you fully discharge the capacitor.

First of all my recommendation would be don't do it, supercapacitors are super-useful in the right applications, yours is not it. If you need your system to stay powered without a constant power source then I would recommend you instead use a small battery, like everybody else.

If however you really want to use a super-capacitor then you need to know the following parameters;

V_max (the highest supply voltage that your system can accept) let's say for the purpose of this example that V_max = 5V4.

V_min (the minimum supply voltage that your system can run off) let's say for the purpose of this example that V_min = 2V4, which matches the minimum voltage requirement of many 3V3 systems.

C_supercap (the capacitance in farads of the super-capacitor) let's say for the purpose of this example that C_supercap = 250F

Now first we calculate the useful voltage drop across the battery as;
V_useful = V_max - V_min = 5V4 - 2V4 = 3V0

Then we calculate the useful charge stored in the super-cap in Coulombs [C] as a function of the voltage difference as;
Q_useful = V_useful * C_supercap = 3V0 * 250F = 750 C

Now we calculate the total power that can be retrieved from the super-cap over the discharge of 750C from 5V4 to 2V4. Here you have to appreciate the fact that the average voltage provided by the super-cap while discharging from 5V4 to 2V4 is going to be the average voltage in between the two, first, we calculate the average voltage provided by the super-cap while discharging as;
V_avg = (V_max + V_min) / 2 = (5V4 + 2V4) / 2 = 7V8 / 2 = 3V9

Now that we know the average voltage output and the total charge difference we can calculate the total energy output of the super-cap in joule [J] as;
E_useful = V_avg * Q_useful = 3V0 * 750C = 2250J

So after analyzing the super-cap and doing some math we now know that we can get a total of 2250J of energy out of the super-cap. Now we need to calculate how long it is going to take the IC to consume those 2250J of energy, this is more tricky as we don't know the exact power consumption of the IC at any one point/ at any specific voltage.

The best we can do given the data you provided is to calculate the minimum and maximum amount of time that the system can stay powered. To do this we first calculate the maximum power consumption of the IC as;
P_max = V_max * I_max
where I_max = 150mA
P_max = V_max * I_max = 5V4 * 150mA = 0.81W = 0.81 J/S (Joule pr. second)

Assuming a maximum power consumption of 0.81 J/S (W) we can calculate the minimum time that the system can stay powered as;
T_on_min = E_useful / P_max = 2250J / 0.81J/S = 2778s (seconds)

So to get this in hours;
T_on_min(hours) = T_on / 3600s = 0.77h (hours)

Now let's calculate the maximum amount of time the IC can stay powered using the minimum supply voltage and maximum supply current as;
P_min = V_min * I_max = 2V4 * 150mA = 0.36W = 0.36J/S
T_on_max = E_useful / P_min = 2250J / 0.36J/S = 6250s
T_on_max(hours) = T_on_max / 3600s = 1.74h (hours)

So likely your IC will stay powered between 0.77 and 1.74 h while not in standby mode.

Now to calculate the minimum powered up time when in standby mode;
P_stdby_max = 5V4 * 75uA = 0.004W = 0.004J/S
P_stdby_min = 2V4 * 75uA = 0.002W = 0.002J/S
T_on_stdby_min = E_useful / P_stdby_max = 2250J / 0.004J/S = 562500s = 156.25h = 6.5days
T_on_stdby_max = E_useful / P_stdby_min = 2250J / 0.002J/S = 1125000s = 312.5h = 13days
OBS!: The above assumes that the system starts out fully charged and discharges in standby mode the whole time.

I have left out the internal resistance of the super-cap which is going to cause an additional power loss and I have also left out the internal leakage resistance which is going to cause the super-cap to discharge by itself without you even drawing power from it.

PLEASE NOTE THAT:
Most MCU's need a regulated power supply and can not run on everything from 2V4 to 5V4 continuously. Probably you are going to need to put a 3V3 LDO between the batteries and the MCU, if you do this then the time the system can stay powered for is much much less:

Let's assume a 3v3 LDO with a minimum voltage drop across it of 1v2

5V4 - (3v3 + 1v2) = 0v9
0v9 * 250F = 225C
225C * (5V4 + (3v3 + 1v2))/2 = 1114J
(5V4 + (3v3 + 1v2))/2 * 150mA = 0.74J/S
1114J / 0.74J/S = 1500s = 0.4h

This is a much more realistic calculation.