if i was born in the usa but had a grandparent from the uk could i move there to stay?

No, citizenship only descends through one generation from parent to child in the UK - your parent was eligible for citizenship but you aren't

You also aren't eligible for the Ancestry Visa because the US is not a Commonwealth country

Very few countries allow citizenship to descend twice from parent to child & from grandparent to grandchild - if your grandparent was born in Northern Ireland that would be much more useful as you could potentially have a claim to Irish citizenship

the fairway card lottery is a variety lottery. Brits are excluded because of the fact there are already many Brits in the U. S.. i particularly doubt you will get into the lottery while you're Irish. Now in case you have been from South Africa, it particularly is a diverse tale.

No ...