There are a few options. I'll assume a Niven-type ringworld with a radius of 1AU (i.e. the surface is the same distance from the sun as the Earth's surface).
If a ring this size were simply in orbit, rotating about the sun once per year, then its occupants wouldn't experience weight. So it actually rotates 41.6 times faster (once every 8.8 days), creating a centripetal force of 1g.
The following ignores the mass of the ring itself (it's made of carbon fiber or something), but at least the first two scenarios could probably be adjusted to account for that.
The "moon" (actually a planet) orbits the sun in a highly eccentric anecliptic orbit, meaning that its orbital plane M is at 90 degrees or so to the ringworld's orbital plane R (figure A). If you fix your camera to a point on the ringworld, the moon follows a helical path around the ring, as requested (figure B). However, this isn't really what you want, because most of the time it would be too far away to see, and when you could see it, it would be rapidly falling toward the horizon and then rapidly rising out of view on the other side of the ring.
You'd never see it pass right overhead. But you could time the orbits so that people in a given town only see it once every 5 or 10 or 50 years.
Orbits like this aren't seen much in nature, for a few reasons, but it's nothing that a ringworld engineer couldn't easily fix.
You have two (or more) moons orbiting each other in a "rosette" (Niven actually mentions this in Ringworld although his terminology is a bit wrong). The center of this system orbits the sun in a circular orbit of the exact size of the ringworld itself, so it looks as if the moons are orbiting the ring (figure C). The center point of the rosette goes round the sun once per year, while a point on the ring goes round the sun 42 times per year, so again, relative to the ring, the moons' path describes a helix (figure D). You will pass under the moons once every 9 days.
This one is a bit harder to visualize. First, let's forget for a moment that the ring is a centrifuge, and assume it's simply floating in orbit round the sun. If you have a single moon orbiting the sun in approximately the same orbit (figure E), then it's in geosynchronous (halosynchronous?) orbit and will appear stationary in the ring's sky.
Now, make that moon's orbit slightly eccentric, so that the apogee (strictly, the "aphelion") is outside the ring, while the perigee ("perihelion") is inside the ring (figure F). It's still sort of stationary, but it's bobbing above and below the ring – if you plot it out, its path looks like this:
It's not exactly a circle, but it does rise on one side of the ring and set on the other.
Now if you recall that the ring is spinning at an extra 40.6 revolutions per year, that shape becomes a distorted helix – I won't try to draw it, but it's the same type of situation as figures C and D.
If it's important to have the moon pass overhead, then you want scenario 2 or 3; if it's more important to have the moon appear at long intervals, you want scenario 1. I don't think there's a way to have both using orbital mechanics, given that the moon needs to get close enough to actually be visible. If none of these options work, then perhaps instead of a real moon you could have a giant airship, or some sort of moon-shaped machine attached to the ring.