Being a circularly symmetric potential, the component of (orbital) angular momentum parallel to the axis of rotation will be a conserved quantity. The periodic boundary conditions give that that angular momentum has a quantum number associated to it as well, though the state of a wavefunction in that region could be a superposition of multiple orbital angular momentum states.
Quantum field theory might paint a different picture, but in ordinary quantum mechanics, the spin state of a particle would not be constrained by a standard potential well unless that potential explicitly depended on spin (instead of just position). As such, the spin is an entirely independent degree of freedom to the situation that the electron is in.
However, if you have two or more fermions, the overall state has to be antisymmetric under a swapping of particles. This does put some constraints on the joint spin state given the joint position wavefunction. In particular, if the position wavefunction is symmetric, the joint spin state will have to be antisymmetric for the total state to be antisymmetric.