Unless you pay to have a true faux finish applied, painting them a single color will never look like aged copper. I've done exterior faux finish work that has held up over time but nothing exposed to the elements like gutters and down spouts. Paint them to blend in.

Much depends on their material - are they vinyl, steel or aluminum? Each has varying success rates for the paint sticking over time when out in the elements. You want something that ages aka peels gracefully.

That said, I have actual copper gutters and the patina color would add a gorgeous accent to your house, if it can be replicated.

If it were me, I would go either real copper or paint to blend in. I usually shy away from gutters as an architectural accent, although on some houses it can work. We painted our brick/siding combination gutters to blend in - horizontal gutters were painted the trim color (off white) and vertical gutters were painted the house color (taupe). They disappear fairly well - and the paint has worn really well (3+ years in a midwest climate).

I would stay away from doing anything that even looked like real copper, but that's because copper thieves are TERRIBLE in our area. Even painting them to look like copper would guarantee I'd come home to gutters ripped off the house.