Forgive me if I make a few assumptions, but it seems that this ring is rather large, considering that its able to completely block sunlight from hitting the earth. I'm also gonna assume that there are gaps or open areas in the ring as its way of switching between night and day on the light side of the planet.
Luckily for the people on that alternate Earth, people have already worked out if something like this is possible, with a few difference. As a matter of fact, its been something thought about doing to a planet in this solar system. It's the sweet young lady next door named Venus!
While instead of using a ring, its been theorized that a giant sunshade would be placed between Venus and the sun, blocking the light and cooling it down. The sunshade could then orbit Venus, allowing the light side face to experience day and night. This ring world of yours could do the exact same thing, although the cost to build it would be a bit larger.
However, why stop on just the side facing the sun? Those poor chaps living on the back side of this Earth want a little sunlight too! So how about we toss some big reflectors extending from the top and bottom of the inner part of the ring so that when the light can be bounced to dark side of the planet, which can be used to give both sides of the planet a day and night cycle.
Or if that doesn't work, then while a ringworld without any gaps in it blocks out the sunlight, a large reflector could be placed in a polar orbit and could provide the Earth with sunlight. Although, neat enough, this would cause the planet's light source to rise and set in the north and south instead of east and west.
If it's all done right, it should be possible to limit any major effects on the weather. I would assume, at least.
Here is a link to the paper detailing plans to terraform Venus, which includes the idea of sunshades, reflectors, and more.
Hope I helped!