In theory it's possible to make a carbon-neutral methane rocket based on atmospheric CO2, though that depends on how completely the methane can be burned.
There are a million viable (and often quite fun) answers here, but one is really kind of funny. What do you get when you mix oxygen and hydrogen? Water? No, of course not! You get rocket fuel! Seriously. Liquid oxygen + liquid hydrogen is a common, and highly effective, fuel that's been used for various engines such as on the Space Shuttle Main Engine.
Rockets can also be carbon negative in another way. A rocket that uses less than 50% of its fuel getting to orbit would be carbon negative, because it's spending less than 'x/2' fuel to go burn at least 'x/2' fuel away from Earth. Factor in some of the fuel coming from carbon neutral sources, and it quickly becomes quite easy for a rocket to be carbon negative.
The flow of mass would be the other way around. Lunar regolith has every element embedded in it and there’s little need to bring anything but bootstrapping -to- the moon. Return to earth from the moon wouldn’t be CO2 producing anywhere but the moon.
Edit: falcon 9, not starship