Anywhere from ancient times to the present.
Question 1: Is there anything there? (answered pretty quickly)
Sunspots, for instance, were first observed by the Chinese in 364 B.C., two millenia before telescopes or camera obscura were first used to study the Sun! While I don't know what the Chinese used, you can see sunspots using a very basic solar telescope. It projects an image of the Sun onto a surface; sunspots - if they're big enough - show up as small black dots. You can actually track their motion, and that of the Sun. It's quite simple, and it's fun. I've enjoyed doing it.
A simple solar telescope, sort of like the one I've used.
I would assume that a Dyson ring would also show up on such a telescope (although I can't test it out). It should dim the star in certain places, showing up as either a black ring or a series of small dots. Assuming they're large enough, and the ring's thick enough, you can see them - and all just using a pretty simple instrument.
Also, consider KIC 8462852, also known as Tabby's Star. A couple years ago, Tabby's Star made some headlines because of unexplained dimming. Its light curve didn't match any of the patterns you'd expect to see from an F-type star; the dimming was large-scale and highly irregular.
Some people thought that the cause was aliens, trying to signal something, and that's what made headlines. What I remember Tabby's Star for, though, are the other hypotheses explaining the dimming:
- There's dust surrounding it, like a debris disk.
- A cloud of comets exists close to the star.
- There are a bunch of small bodies - maybe asteroids - around it.
The last two scenarios are similar to what you're talking about here. Assuming that KIC 8462852 has dimmed because of a cloud of bodies like this, we can detect such a disturbance from over 1,000 light-years away.
With fluctuations of that magnitude (~22%) - and I'd assume a Dyson ring would cause such fluctuations - the civilization could figure out that something was there as soon as observations of their parent star started.
Question 2: What is it? (answered pretty slowly)
So, there are objects around the star. That much can be determined. But what are they? Are they natural? Are they artificial? If so, who made them? How do they work?
Think back to KIC 8462852. We thought up a whole bunch of things that could be causing the dimming. All of them seem realistic. But would any of them have occurred to a primitive civilization? Remember that early astronomers didn't know much about our Solar System. For instance, it took Galileo to figure out that Jupiter had moons, and Huygens to determine that the bulges around Saturn were rings. Planetary rings weren't known about by the early Chinese. Comets were observed, but nobody knew their true nature.
So, they'd really be stuck for ideas. I guarantee you that they wouldn't come up with the idea of a Dyson ring - going to space would be pretty far-fetched. Remember, they might not even have a heliocentric model. However, if you fast-forward a couple millenia to a civilization like our own - one that's sent probes across the Solar System, to study the planets, the Sun, and many other bodies - then you'd have find a group of astronomers who can figure it out.
I'm going to posit that on Earth, someone would have figured this out by ~1900 (i.e. pre-space travel), maybe a bit later. But acceptance of the idea would depend on a few things:
- Are people willing to study these objects deliberately, or are they just considered minor and unimportant?
- Do the people believe aliens could exist?
- Do they know that such a method if even possible?
Figuring out these societal beliefs can give you a better idea of the time frame you're looking for. It will be dependent on telescopic development.