What is the value of an 1801 dollar gold coin?

The US didn't mint any $1 gold coins in 1801. My guess is that you almost certainly have a modern presidential dollar coin. 1801 is the date of the start or end of that president's term - the minting date is on the coin's edge. And it's made of brass rather than gold so it's only worth $1.

Feel free to spend it. Hundreds of millions are being minted.