Dell's motherboard(s) of that era actually state the max power available through the PCIe slot right on the board. For example:
You'd have to double-check on your board specifically.
There's no guarantee of functionality, nor stability if you use an underpowered slot. One things for sure, even if it'll run, the card will never be stable under load as it's not fully powered.
Replacing the PSU will not change the limitation (if applicable) from the motherboard. A card with external PCIe power, paired with a replacement PSU should work though.
As for the "upgrade" part, which 790 do you have specifically? I believe those can in everything from a full mid-tower, to an ultra SFF rig.