So while I am still interested in finding better answers, I want to share what I ended up settling on for my own world and everyone else's reference. Overthinking? Maybe. But nice cities are my kind of worldbuilding.
tldr; cities would choose to build in the "iris" of the eyeball world.
Quick definitions and an (approximate) equation
Here's our situation, assuming a spherical planet for simplicity:
The sun being fixed is inconvenient in that we need to plan around it, but convenient in that it makes calculations easy. At zenith (or the substellar point directly under it), the angle between the sun and ground is 90 degrees. At the terminator or day/night boundary, it is on the horizon and the angle is 0 degrees.
The size of a shadow is just: $H/\tan({\theta}) = L$. Where H is the height of the building, $\theta$ is the angle of the sun, and L the length of the shadow.
Rejiggling slightly we can get an equation we can use for city planning:
$$(AverageFloorHeight*NumberOfFloors)/\tan({\theta}) = AcceptableShadowLength$$
Obviously your city needs to make some judgement calls on what is acceptable. And may require/provide forms of illumination to cover the shadows at street level.
Architecturally perhaps, building on @Krišjānis Liepiņš's suggestion, buildings are intentionally build to reflect light onto street level. On the flipside, building off @JBH, the sunward sides of builds may feature blackout curtains for when people want sleep.
However regardless of what you pick the overall trends are the same unless you don't care about this at all.
Also you might not want to have roads running directly into the sun, especially at low angles of elevation. Just so you don't blind people going in one direction. Less of an issue with trains though.
At the extremes
At the zenith/substellar point
Directly under the zenith the shadow is basically zero, so on these grounds you can build as tall as you want. However, the substellar point is ... a point. Whatever geography is or isn't there, is what you get. Also the interiors of the buildings would receive little sunlight. Overall this may not be ideal.
At the terminator line
You can't build anything on flat ground without casting an impractically long shadow. Anything built here would likely be picturesque, but low density, "twilight towns" built on/into things like mountain slopes. But not material for large cities, these would be restrained by geography. It would be a potential boon to nature though if large portions of the planet are considered impractical for urban development.
So if neither are ideal, instead consider...
"Iris cities"
Which is what I'm terming them. You can develop within the radius determined by how tall you can build practically while keeping the shadows cast at an acceptable length. For instance, say you decide your floors are 4.5 meters tall at most, you find a shadow length of 3 meters acceptable (maybe your streets are 9 meters wide, and 33% shadow is acceptable), and you want to be able at least build 5 story tall buildings. Well you need to build between an angle of elevation of 82.41 minimum to 90 maximum. Which means very close but not necessarily on, the substellar point.
So you get,
- Low-rise cities only! Which we do have on Earth. Skyscrapers would be even more obnoxious and impractical. Luckily low-rises are pretty efficient from an infrastructure standpoint, even here on Earth.
- High population. Urban development elsewhere is far less practical. So there would be a lot less cities to "compete" with for residents. And for that reason, people would find housing and other amenities much easier to come by within the "Iris".
- A lot of free solar and wind energy. You won't have issues with energy generation.
- Agricultural production may also be able to take advantage of the constant sunlight. Plants need time out of the sun as well, at least on Earth.
- Storms: The worst weather the planet can throw at you is likely to occur here. Cities are going to want to take advantage of geographically sheltered areas, and have elaborate defenses against extreme weather. They can expect to take a hit from time to time.
Conclusion
The sun may actually have significant effects on development. Barring everything becoming extreme low density housing or a dystopian shadowed cities, there would actually be significant factors pushing towards concentrating development/urbanizing a relatively small area.
So you may end up with low-rise mega-cities being the norm. Considering that kind of development is both efficient for the city, can lead to affordable housing with an ample housing supply, and leaves the rest of the planet to the wildlife ... it could actually become quite nice indeed.