Integration on the punctured plane

You did not obtain answers so far, because your description of the problem is somehow overloaded. The underlying domain of integration is an "abstract copy of $S^1$". This copy can be realized as ${mathbb R}/(2pi{mathbb Z})$ or as geometric unit circle $partial D$ in ${mathbb R}^2$. At any rate the chosen parameter variable is $theta$, and for a full performance through $S^1$ this $theta$ should run over the interval $[0,2pi]$. On the other hand it is true that this $theta$ is not a 100% coordinate function on $S^1$, but we have control over the details $ldots$

Understanding all this, your $f$, defined on the points of $partial D$, should be considered as a function $$rho(theta):=f(e^{itheta})in ]0,1[qquad(0leqthetaleq2pi) .$$ At any rate, your $j^*alpha=dtheta$ is correct; and $$int_{cal C} j^*alpha=int_0^{2pi}dtheta=2pi .$$ There is no question of an integral $int_0^1dtheta$ here.

Maybe it is of some help to note that $alpha=darg$, since $$nablaarg(x,y)quadleft(=nablaarctan{yover x}right)quad=left({-yover x^2+y^2}, >{xover x^2+y^2}right)$$ is valid for all $(x,y)ne(0,0)$.