The 7750 should run on the power supply in that small form PC. Only catch may be that some OEM systems do not put full power through the PICe and may have some other funny issue, but that is not all system and not with all video cards.
" Kinda hard for an IT enthusiast that knows just enough to sound stupid", don't worry about that, you sound stupid in a very smart way
Just keep away from small, hard to upgrade systems in the future LOL
Also make sure you read that 7750 specs and how it works, it has all those adapters, make sure the adapters it comes with or you can buy for it match your monitor inputs.