Not seeing the shielding claimed from "space radiation" if the thing looks like the illustration. "Space radiation" comes from all directions, so the big open ends of that spinning tube are tremendous holes in the "shield".
And, if the asteroid material is so easily displaced that spinning it to 1 G of centrifugal force will make it fly out to the mech tube, then building structures "on top" of it (i.e., at a slghtly smaller radius from the center), and counting on it to support those structures against radial force would seem like building on an unconsolidated rubble pile here on Earth - not acceptably stable.
And, then there is the need for propellant to accelerate the spin. And of course the need for energy to run systems - so where are the solar collectors and how do you point them at the sun on a rapidly spinning tube - maybe just coat the whole exterior with solar cells?
Finally, how stable are carbon nanotubes against the space environment? If that mesh tube has a failure, this whole city would have a disaster.