= Solution
The <spin> operators here are <dimensionless>: $S^j=\sigma_j/2$, so $\Omega$ has units of inverse <time> and $\Phi$ is a <dimensionless> pulse area. Integrating the Schrödinger evolution across an infinitesimally short pulse gives $U_\Phi=e^{-i\Phi S^x}$. The finite free <Hamiltonian> contributes a vanishing integrated phase within that pulse. Between pulses the free evolution is $U_\Omega=e^{-i\Omega TS^z}$.
In the $S^z$ eigenbasis, $S^z=\operatorname{diag}(1,-1)/2$ and $(\sigma_x)^2=I$. Consequently
$$
U_\Omega=\operatorname{diag}(e^{-i\Omega T/2},e^{i\Omega T/2}),\qquad
U_\Phi=I\cos(\Phi/2)-i\sigma_x\sin(\Phi/2).
$$
Starting just before a pulse, the kick acts first and the free interval second. Operators act on states right to left, hence
$$
\boxed{U(NT-0^+,-0^+)=(U_\Omega U_\Phi)^N.}
$$
There are exactly $N$ kicks, at <times> $0,T,\ldots,(N-1)T$; the pulse at $NT$ is excluded. Sampling just after kicks would change the displayed product order. This endpoint convention specifies the <delta-kicked spin Floquet operator>.
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