If we used a conservative 3.5% average return to give $1,000 USD for each American (~325M) you need nearly 10 Trillion dollars in this fund or over half of our yearly GDP.
Now keeping in mind this would need to be generated from a tax on some good/service/capital which obviously has a cost component at it's base. Just getting the initial 10Trillion in a fund would be ridiculously hard. Now add to that the fact that 1k is a pittance and figuring out a true basic income -- this is wayyyy more complex.
I'd like to point out a HUGE fallacy in this "simple" plan -- it only works in low population high natural capital resource areas. Alaska and Norway happen to be relatively remote/cold places with an abundance natural gas/oil. Norway's 1T dollar fund would be comical at US scale.