For any star to always be "locked in place" to a construction (some stonehengy-like-thingy) would require the planet to be not turning and not orbiting. If it turns at all, then the entire globe of the cosmos is revealed in one rotation (or, more accurately, the entire hemisphere of the cosmos, unless one is standing on the equator). And if it orbits at all (using a 2D example to make the point), then there is a time each year when the sun is between the star and the construction. No combination of orbit or construction will circumvent this.
Worse, the idea that the planet is rotating at the walking speed of your nomads — which might have significant consequences for its magnetosphere — doesn't solve the problem of the orbital speed, which must meet certain mathematical realities or the planet can't stay in its orbit.
But that's really boring.
First of all, your people are nomads, constantly moving. So the question can't be whether or not a star can be locked to just one construction, because the nomads will only be at that construction once-per-really-long-and-enhausting-journey-around-the-planet.
But what if there's a series of constructions. Could the same star be seen at each construction if the nomads are using it at the proper time?
Well... No, but...
We're back to the problem of rotation and orbits... unless I consider a star like our own North Star, Polaris, and if we stick to your basic chart, then an equivalent star would be visible in the night sky all year round. The closer said star is to the axis of the planet's rotation, the more it will be in the same spot every night all night long (assuming their path around the world is more-or-less along the same latitude).
So, yes, if you're willing to live with some basic astronomic realities...
A north star would work perfectly for any construction designed to refer to it anywhere along your nomads' path. Not surprisingly, the construction, like a compass, will always point north. In that regard, a compass would be much simpler than a construction — but you didn't mention the tech level of your nomads. This solution will work regardless the speed of rotation or the position of the planet in its orbit.
A "north star" that was a little off-axis would be more fun because the constructions would need to be more intelligently placed. They would not always point north. This solution would still be independent of the planet's orbit, but would be dependent on its rotation, meaning if you can't see the star through the construct, you're either too early or too late (early is recoverable, late is bad...).
However, the further off-axis the "north star" is, the more dependent its visibility will be on the orbit of the planet.
But, to make a point...
Why are you working with just one star? Humanity started giving names to constellations pretty much as soon as it learned to draw with a burnt stick. The repeatability of those constellations through the course of a year has been constant for all of recorded history. Thus, your problem ceases to exist if you permit more than one star (aka, constellations) to guide your nomads.
We have arrived in the valley of Carook! The fields are lush with the grain we planted when last we were here and the forest is full of game! Send L'due and N'yela to stand at the Star Stones to see if the toes of the Leaping Jester are at least twenty rotations of the sand glass from standing atop the stones. We must leave the moment the Jester stands upon the stones!