Hi,
No browning if you want to be historically correct. It was becoming popular in England during the last decades of the 18th century but did not catch on in America until the 19th century. Rust bluing would also not be historically consistent for a 1790s Rupp. That said, eventually unfinished iron barrels turned brown with rust. You might try tarnishing the barrel with a light browning or use quick bluing, blue the barrel, and then rub it back to a light gray with Scotch Bright pads dipped in water. Here are some examples:
However, if you intend to do serious target shooting, I would brown the barrel even though it is not quite correct for the period. I urge you not to leave the lock with the textured cast finish and brown it thinking that would look authentic. It doesn't. Polish the lock plate, ****, and frizzen to get rid of the cast surface and leave it bright or temper blue it. Here are examples.
dave