= Solution
Write the <exact differential form> as $\alpha=du$ for a smooth real-valued function $u$ on $N$. On $T^*N$ take the <Hamiltonian function> $H=u\circ\pi$. With $\omega=\sum_jdq_j\wedge dp_j$ and $\iota_{X_H}\omega=-dH$, the <Hamiltonian vector field> is
$$
X_H=\sum_j\frac{\partial u}{\partial q_j}\frac{\partial}{\partial p_j}.
$$
Its complete flow is explicit:
$$
\boxed{\phi_H^t(q,p)=(q,p+t\,du_q)}.
$$
The base point stays fixed, so this flow exists for all real $t$, even if $u$ is unbounded on a noncompact base. Its time-one map is exactly $f$, proving that the <cotangent fiber translation> is a <Hamiltonian diffeomorphism> in the unrestricted complete-flow group.
This construction also justifies the hint for this family, without treating exactness as a general substitute for a Hamiltonian isotopy. For any <Hamiltonian isotopy> $\psi_t$ on this cotangent bundle, <Cartan's magic formula>, $d\theta=-\omega$ and $\iota_{X_{H_t}}\omega=-dH_t$ give
$$
\frac d{dt}\psi_t^*\theta=\psi_t^*\bigl(d(\theta(X_{H_t}))+\iota_{X_{H_t}}d\theta\bigr)=d\bigl(\psi_t^*(\theta(X_{H_t})+H_t)\bigr).
$$
Integrating proves that $\psi_1^*\theta-\theta$ is exact. For the present translation this difference is $\pi^*\alpha$, which is exact if and only if $\alpha$ is exact: one direction is pullback of $du$, and the other is pullback by the zero section. Necessity therefore follows from the differential identity, while sufficiency follows from the complete flow just constructed.
Compact support is a separate issue. If $du$ is nonzero at some $q$, the map moves every point of the entire unbounded fiber over $q$, so it is not a member of the <compactly supported Hamiltonian diffeomorphism group>. For example $N=\mathbb R$, $u(q)=q$ gives $(q,p)\mapsto(q,p+1)$. Thus the claim about this global translation uses the full Hamiltonian group, not $\operatorname{Ham}_c$; its flow can be cut off to agree on a prescribed compact swept set, but such a cutoff does not give the same map on the entire cotangent bundle.
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