OK, so now the caveats.
First of all, you need to ensure your program will finish the EEPROM write prior the "big cap" will be drained. This might or might not be trivial, depending on the application and its complexity. You need also consider that the actual charge you can use is given by the difference of voltage on the fully charged cap and the minimum voltage needed for the regulator (btw. 7905 is a negative regulator, and 7805 is really power hungry, so maybe not the best option for this application), so your AC source should provide high voltage enough and the voltage regulator will need to drop that voltage (with the consequence of a power loss+heating etc.) during normal operation.

Also, you need to make sure the mcu won't attempt to write into EEPROM until the big cap is fully charged - most of the time this is fulfilled but with short pulses on the AC mains you might get into trouble.

Then, if you calculate the needed capacity of the "big cap", take into account that electrolytics have quite high tolerances, and worse, they lose capacity and develop leaks when aging (especially when stressed by heat and/or current pulses). I know nothing worse than an application that develops spurious problems
after a couple of years of operation. Although I know little on supercaps, I wouldn't believe too much in their long-term stability, too. So we are probably left with tantallums on this position, with somewhat limited capacities and voltages.

An another thing, you need to make sure that the peripherals won't "suck out" the "big cap" too early - even if they are not connected directly to the . Beware, this behaviour can/will change depending on their state and the connected "outer word".

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I did not say this can not be done, as I don't say the IRAM backup cannot be done (in powerdown mode). My point is, that each of these is not completely trivial and you need to consider carefully all the circumstances of your particular application.

Jan Waclawek