The question is tagged hard-science. I'm going to ignore that tag completely. I think it was bad advice to add it. Here's why:
Understanding the sun-climate connection requires a breadth of expertise in fields such as plasma physics, solar activity, atmospheric chemistry and fluid dynamics, energetic particle physics, and even terrestrial history. No single researcher has the full range of knowledge required to solve the problem. To make progress, the NRC had to assemble dozens of experts from many fields at a single workshop. — From "Solar Variability and Terrestrial Climate," 2013, emphasis added.
In other words, if we tag this question hard-science, delete my answer, and wait for one person to provide an insightful answer, the question might be the only question on WB:SE to qualify for the Tumbleweed badge.
On the other hand, in a rare instance where no single question may represent the "one best answer" Stack Exchange prefers to see, the sum of several insightful answers may provide the OP what he's looking for. I'm moving forward on that premise.
To the Peanut Gallery: If my science is off, rather than whining about it in a comment, add an answer that explains why my science is off! This is a complex enough subject that your insight is infinitely more valuable than your complaint as it will take a lot of us to provide a "complete" answer.
Solar radiation causes pressure zones
While certainly not the only contributor to atmospheric pressure, the sun is a major contributor. As atmosphere heats, the molecules spread apart (basic gas dynamics), pushing on surrounding atmosphere. Generally speaking, on the ubiquitous sphereical planet having no land, no water, and just a uniform blanket of atmosphere, the lowest pressure point would be directly under the sun (the point where the molecules are spread furthest apart due to heating) and the highest pressure points would be close to the horizon of the sphere where warmed atmosphere is pressing against the cool atmosphere from the backside of the planet. If the planet didn't rotate and the sun didn't orbit, then this condition would stabilize into a static condition.
But we do have rotation and orbit (and, just for fun, luminosity variation). Let's assume our test sphere doesn't rotate, but the heat source (our "sun") does. This would cause the pressure zones to move around the planet with the sun. The leading edge is fairly simple: you get wind in the direction of the orbit as the high pressure ridge is pushed into the cooler air, warming it.
The trailing high pressure ridge is more complex. Yes, it sucks the air along in the direction of the orbit. But the "drag" as it goes causes turbulence and eddies. (Watching a wind-tunnel test of an inefficient vehicle design will demonstrate this behavior.)
Let's add land & water
Let's mess up our lovely spherical world by adding land and water. We could ignore the heating effect on the sphere because everything was uniform. Thus, while it would complicate the model a bit, it wasn't enough to worry about it.
However, land and water act very differently. Land at the surface heats quickly because it is generally a bad conductor of heat. Water heats slowly due to its high thermal conductivity, but when it does heat, it adds water to the atmosphere.
Land
Land messes up our beautiful wind by creating local high pressure zones. This is one of the reasons why you see canyon winds. Different kinds of land heat and cool at different rates. Different heights of land (valleys, mountains, etc.) channel wind and add their complexity. If we ignore water, the general affect of land is to cause wind to change direction.
Water
But water... we love water... By itself water doesn't significantly change the direction of wind. This is why winds such as the Prevailing Westerlies exist and why they are so valuable to transportation. It's evaporation that causes the greatest change. Evaporation creates local cool areas and saturates the atmosphere with water vapor. This permits storms.
Simplifying the system outrageously, storms happen when:
There's too much water in the atmosphere, causing drops to form that are heavy enough to fall,
When something gets in the way, causing the water to "bunch up" (think "high pressure zone") and form drops (mountains are great at this), and
When there's turbulence.
That trailing edge we talked about earlier can be thought of as a massive storm in the making. All it needs is water and you get all kinds of things, from gentle rain to hurricanes.
The leading edge is more complex because the wind it creates is fairly consistent, so it's things getting in the way (like mountains) that tend to cause rain.
Please note that I'm not even going to try to talk through the effect of oceanic currents and rivers on climate. Not only is it out of scope for the question, but they add an unbelievable amount of complexity. However, if you're getting my jive about heating/cooling/evaportation, etc., you can see how to create simple rules for how rivers and currents affect weather.
Finally, the complex rotation, orbit, and luminosity pulse
The chart you refer to represents what an individual would see if they pulled out a barcalounger, some cream soda, and a lot of pizza and simply watched how the light level changes over the 648 hours of the world's "Great Day." The chart's only value in this case is to give us an idea of how the energy piston of the sun is pushing things along.
In my previous examples, the "sun" was creating an effect that could be thought of as moving around the planet with the orbiting sun. From the perspective of an individual, high pressure zones, high pressure ridges, and low pressure zones would come and go over time. If you think of such a simple solution in terms of a sine wave, the low pressure zone is when the sun is directly over you, the high pressure zone is when it's behind the planet, and the high pressure ridges are at the inflection points. As you watch this pass by in time you can (hopefully) see what I mean by "energy piston."
The critical take-away from this visualization-without-a-graphic is that storms generally happen at the inflection points.
Our rogue planet has two sets of inflection points: the major points that occur every 18 hours during the "day" cycle and the minor points that happen once every 324 hours during the "great day" cycle. The confluence of great-day and day inflection points have the highest liklihood of whomping big storms. However, this perspective is only 100% true when we consider the sphere with uniform atmosphere. In the more complex picture, they will add greatly to the weather mess.
Conclusion
I believe this planet will have very strong "westerlies." Much stronger than we would experience on Earth. I believe its potential for storms is greater, too. However, given the circumstances, there will be no seasons and no seasonal change to the polar caps.
Dude, weather is nowhere near that simple
As a final remark, my simplification is extreme. Here on Earth, whole days can pass without any wind or significant change in weather (other than at canyon mouths... that's actually pretty predictable). Why didn't I try to explain all this? Well... (a) I don't know a thing about the rogue planet's geography, (b) I don't know a thing about the rogue planet's hydrology, (c) I'm ignoring some whomping big cause-effect sources including but not limited to rivers, ocean currents, volcanoes, and polar caps, and (d) I'm NOT a climatologist, nor an expert in any of the fields listed in the above quote. If you're expecting a thorough answer, you'll need to join the process. Cheers.
