Given that a rigid Dyson structure doesn't work by known science, who knows?... but probably not.
To make a stable, rigid Dyson sphere or ringworld work around a star you need not one, but MULTIPLE sci-fi engineering elements.
- How does it remain stable around the star?
- How is there gravity on the surface?
- (A ring special) How is the atmosphere kept from falling over the edge?
- How does it keep from collapsing on itself?
- Where do we get the material?
If we don't know what it's made of, what's holding it together, what's holding it in position, or how things are held onto it, then it's so divorced from reality that any prediction about how long it will last without maintenance is as correct as any other. Maybe with some of those holes filled in a prediction could be made, try another question with more detail?
Then there's the second part of the question: how long does it take to evolve an intelligent species from the greatly culled gene pool that's left in future humanity's zoos? Again, the answer is who knows?... but probably 10 to 100 million years if it happens at all.
We only have one example of an intelligent life form evolving naturally, humans, so it's hard to extrapolate from a single data point.
Depending on where you start you get a number anywhere from the evolution of primates 85 million years ago to Homininae 14 million years ago to Homo 2.8 million years ago.
There's nothing about evolution that says it leads to intelligence. Intelligence might be a fluke, a terrible survival strategy, and evolution is done with that. In that case the answer is "never".
Another way to approach the question is from an engineering/cost POV.
Engineering is a trade off between features and cost/time. Given the immense difficulties of building a rigid ringworld, why would the engineers add to the problem? While automatic maintenance and repair systems will surely exist, why would they over-engineer them to work completely autonomously for millions of years? There's no human need for this.
As an example, surviving ancient structures have lasted not because the builders wished them to, but because they are very simple, their materials are very durable, they have no moving parts, and they got lucky. A pyramid is the shape a pile of stone naturally makes. Most everything else they made is gone, we see what little has survived. Even then, they've only had to last a few thousand years. In a million years all of humanity's structures will be gone.
Which is why I say the answer is, even knowing nothing about their technology: probably not.
Humanity would more realistically create a Dyson swarm which, if they got the orbital mechanics right, would last pretty much indefinitely.
Here is my "clever" answer: immediately. The new life is the AI necessary to maintain the ring.
Many people have been arguing in the comments that it will have auto-repair and maintenance technology (agreed) which will, unattended, keep everything working and hold back nature for the millions of years necessary for life to evolve (nope) even though it was never designed to do so because it was never a design requirement. It would have to somehow improve itself...
The builders made themselves an AI, or near enough that, left unattended, it achieved enough intelligence to adapt and survive. That is the new life which appears on the ringworld.
It's a feasible extension of the design requirements: the designers would need a very smart machine to run the infrastructure. Its AI scoring algorithms would be to maintain the ringworld in a stable configuration. The human builder could even be routinely holding it back from becoming too advanced. Once the human restraints are gone, and with the additional pressure of having to now do everything itself, a ringworld-sized computer could evolve in a matter of minutes once it's allowed to go rampant to achieve its goal.
It could even have wiped out the builders, if you want to go that route.