= Solution
The vertical tangent space of the trivial principal $\mathbb R$-bundle is spanned by $\partial_z$. Since
$$
D=\operatorname{span}\{\partial_\theta+f\partial_z,\ \partial_t+g\partial_z\}
$$
projects isomorphically onto the tangent space of $S^1\times\mathbb R$, it is always complementary to the vertical direction. It is the <horizontal distribution of a principal connection> precisely when it is invariant under the principal translations $z\mapsto z+a$. The horizontal lifts of $\partial_\theta$ and $\partial_t$ are unique, so this invariance is equivalent to
$$
\partial_zf=\partial_zg=0.
$$
The functions must also be smooth and <periodic function>[periodic] in $\theta$, as is already required for them to be functions on the cylinder.
Under these conditions the connection form is
$$
\mathcal A=dz-f\,d\theta-g\,dt.
$$
It sends $\partial_z$ to $1$, is translation-invariant, and has kernel $D$, proving sufficiency as well. Since the structure group $\mathbb R$ is <abelian group>[abelian], the bracket term vanishes and
$$
\boxed{\mathcal F=d\mathcal A=(\partial_tf-\partial_\theta g)\,d\theta\wedge dt.}
$$
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