That inner surface will get more light than you think
Light's a funny thing, especially when you bring an atmosphere into play. There's a number of things that come into play, including atmospheric lensing and atmospheric refraction. Here on Earth, these effects come into play during sunrise and sunset, when light is visible before the sun is physically above the horizon.
What this means for your torroid planet is that the inner surface isn't as dark as you might think, nor as cold. Certainly colder than the planetary face nearest the sun, but not completely dark anywhere (possibly not even at the inner equatorial regions, but this would depend on the inner and outer radii and the nature of your atmosphere). Keep in mind, you have no tilt, which means there will always be light at the poles... the entire circumference of the poles.
So, the real question is, what are you trying to achieve from the perspective of suspension of disbelief?
And this is imporant. After all, we have no evidence of the reality of a torroid planet, so the idea of describing the most realistic fantasy planet possible is somewhat near-sighted. But! Combine this with the fact that astronomers are being surprised yearly with new discoveries in the universe that just a few years ago were believed to be impossible and what you're really looking for is a plausible explanation.
If the inner-R is "small" (whatever that might be) or the atmosphere thin, then I propose it's plausible to assume the inner surface up to the "poles" (the "top" and " bottom" of the planet or the "poles" from the perspective of someone living on the outer equator) would be frozen.
If the inner-R is "large" (whatever that might be) and the atmosphere reasonably Earth-like, then I believe the inner equatorial region could be quite warm. Especially if there's any volcanism in that area. Not necesarily Sahara Desert warm, but high Alpine warm. In this case only the polar circumfrences would be frozen.
The thicker the atmosphere, the warmer the inner equatorial region.
Now, a perfectly reasonable argument is that no matter how much atmospheric lensing and refraction, the inner surface can never be any warmer than the polar circumferences. Good point! But hold that thought while I get through the next section.
Predicting climate is a LOT harder than herding cats
And @WillK is very right about the wind. Climate is complicated to predict to begin with, but adding that inner equator makes the whole thing exponentially worse. You'd have winds rotating with the spin of the planet (like our Prevailing Westerlies), but you'd also have wind wrapping up and over the poles and a vortex around the inner equatorial region.
Add to that some dog-ugly ocean currents. Oh, the ocean currents....
And the result is, as @WillK suggests, an effect that would moderate the cold.
Maybe...
The single biggest problem is that, insofar as we know, habitable torroidal planets cannot exist. Gravity would pull them into a sphere. And if you handwave that, then there's a completely unreasonable extension of the Shell Theorem that would suggest that anyone standing on the inner equator line won't experience gravity and gravity would only increase to "normal" as you approach one of the polar circumferences... But it also means that every possible means of predicting the climate goes right out the window.
Conclusion
And that's the problem with cool worlds that (insofar as we know) don't exist. You're looking for a rational, justifiable answer to a question that's fantastic. I love it! But it means you'll need to accept my half-baked logic as a potential solution — because in the end, there's no scientific way of trivially proving what should be true. Only expressing what could be true.
So, in the end, I hope you take these observations and use them to design an incredibly fantastic planet! And not worry about how scientifically plausible any of this might actually be.
The muse is driving me to add something to this answer — and I'm not going to get any sleep until her demands are met — and I'd really like to get some sleep...
I need to underscore why it's important that you simply choose the conditions you want and move forward.
A terrifically simplistic climate model
Start with a sphere. It's not rotating (certainly not orbiting) and there's a heat source some distance from it that doesn't cause it to burn, just heat up. The sphere has an atmosphere. What does that simplified model do?
The air closest to the heat source heats up, causing a low-pressure zone. It pushes air toward the other side of the sphere opposite the heat source where a high pressure zone develops. A constant "storm" (turbulent wind, no water yet) exists at the terminus where hot and cold air is constantly competing for the same space. Because the air closest the heat source will be hotter than the air near the ground at the same point, you'll have a "rotation" of air from the point along the equator closest the heat source to the terminus in all directions. You'll have something similar on the back side where warmer air rises and cycles away toward the terminus. If you think about it, those cycles (in this incredibly simplistic model, which makes angels weep, by the way), are the basis of the cycles we see here on Earth.
Next, we start the sphere rotating. This rotation does a lot of things (no water, no "landscape" yet, just our perfect spherical horse planet — assume no tilt). It causes the "prevailing westerlies" to develop as the spin of the planet causes the air to start moving along with it. That causes the cycles I earlier mentioned to begin to stretch along the latitudinal lines of the planet. Because the sphere is rotating (a turn/day so to speak) the tangential velocity of a point on the equator is quite a bit faster than the tangential velocity of a point near a pole. This causes those stretched cycles to break up along latitudinal lines (now those cycles are looking a lot like what we see on Earth).
So far so good. Our ridiculously simplistic model makes a kind of sense.1 Next, I'm going to introduce three things that are so complex that trying to describe their effects in this post beyond the summary I'm about to provide is, well, basically impossible. But bear with me, it's all important when we get to your torroidial world.
Add water and think about the rain cycle. Wherever water goes, it sucks up heat. Evaporation cools things down. Condensation (you know, "rain," caused by cooling things down) further cools things down. And when the rain falls on the ground, it cools the ground. The rain cycle creates localized high/low pressure zones and, therefore, localized turbulence. But they also create a more dense atmosphere by adding moisture. Dense atmosphere is yet another "high pressure" zone. The Rain Cycle gets in the way of all those lovely cycles we earlier discussed and really messes things up. BTW, keep the word "moisture" in mind — it's about to become important.
Now add ocean currents. Never underestimate the effect of a good ocean current on the atmosphere. The air literally gets pushed along from the cold currents to the warm currents. If you're thinking, "dang, now I'm starting to understand where hurricanes and cyclones come from," you're starting to get it.
Next, add soil differences. The effect of desert sand on the atmosphere is quite a bit different from the effect of loam or the hardy rock of many (most) mountains. Worse still are the "soils" (you know, "stuff sitting on the ground") known as "vegetation" and "snow." Soil differences also cause local high/low pressure zones and push the air around.
Finally, although not particularly relevant to this discussion (but included for completeness) are the geological specifics of your world. Mountains and canyons and valleys... All these things cause local high/low pressure zones near the ground.
Yeah, yeah, yeah... but what about my torroid?
Here's the problem: Where with a sphere the sun's behavior is to heat the air closest to it and cause that air to "cycle" toward the back side of the planet (using that first simplistic model for convenience), now it's trying to do that with a honking-communist huge hole in the middle of the planet. Some air will get to the back-side of the planet, but some will get trapped in the center of the planet.
And that's important, because not only is that air carrying that hot/cold (low/high) cycle into that center space, it's also carrying moisture. And what's that moisture doing?
Because the center is cooler than the face of the planet closest to the sun, the moisture is condensing... into rain.
Because we're dealing with the Shell Theorem, the air in the inner-equatorial region is spinning in a nasty vortex free of gravity. Kindof... How much "reality" are you willing to give up? You've already given up a lot of reality to have a torroidial planet. If we let the atmosphere have gravity, then the consequence of the rain, probably freezing, spinning around in that vortex near the inner equator... is a growing ball of ice.
But that ice, starting from the infamous $T=0,$ isn't just sitting around in the center. Remember, no planetary gravity at the center.2 So as it spins and gives in to random changes in its shape, weight, density, and size (aka Chaos Theory), it starts bouncing around in there. As it (or pieces) bounce near the polar circumferences, planetary gravity takes over and the chunks settle to the ground.
What's the consequence?
This is the really important part and the reason why you need to decide what the rules of your planet are and run with them rather than worrying about "science." The chunks eventually close over the central hole both at the north and at the south.
Because some chunks are still in that central, near-zero-gravity area, you now have a baby's rattle. The cool part is that this means, upon occasion, a chunk might break through the northern or southern shell and scare the beejeebers out of people.
But the growing weight (OK, think "thickness") of the northern and southern shells will also result in ice slowly encroaching on the inner equatorial region.
Result? We now have a spherical planet with an ice-filled "doughnut hole." Volcanism may occasionally cause instability, but that instability has a long way to go to get to the "surface" near the polar circumferences.
Another result is that the rest of your planet is a desert because, eventually, every drop of moisture will find its way to that central hole where it condenses, becomes ice, and fills an inconceivably big hole.
So... having gone through all this illogical logic... can you see why I'm advocating that you simply choose what your planet's rules will be and move forward?
1 And if you're about to complain about my ridiculously simplified model, or if you think it doesn't make enough sense... you can bite my shiny metal..... I knew that cartoon would come in useful someday.
2 OK, "little to no gravity." Remember, "ridiculously simplified model." Please refer to footnote #1.
