Embedding/Submersion Properties of Cotangent Maps (Pullbacks) Let $M$ and $N$ be smooth manifolds and $f: M \to N$ a smooth map.
Define the pullback bundle $\pi_f^*:f^*(T^*N) \to M$ as usual by
$ f^*(T^*N) = \{(x,j^1_{f(x)}g) \in M \times T^*N \}$ with projection $\pi^*_f(x,j^1_{f(x)}g)=x$ then the pullback of $f$ is a smooth morphism $$f^*: f^*(T^*N) \to T^*M$$ and moreover a vector bundle morphism over the identity defined by $f^*(x,j^1_{f(x)}g) = j^1_{x}(g \circ f)$. Now the questions are:
1.) Suppose $f$ is an embedding. Is $f^*$ a surjective submersion?
2.) Suppose $f$ is a surjective submersion. Is $f^*$ an embedding?
3.) Suppose $f$ is a diffeomorphism. Is $f^*$ a diffeomorphism?
(Regarded $f^*$ just a smooth map, forgetting the additional bundle structure)
4.) Suppose $f$ is a diffeomorphism. Is $f^*$ a vector bundle isomorphism? 
 A: Let me adopt slightly different notation, because otherwise the symbol $f^\ast$ will be overloaded. Fix $f: M \to N$, and let $E$ be the pullback of $T^\ast N$ to $M$. Then there is a natural map $\phi: E \to T^\ast M$ given by $\phi(x, p) = (x, (df)^\ast p)$. All of your questions really come down to: what is the relation between $df$ and $d\phi$? In coordinates we have $\phi = (\phi_1, \phi_2)$ with $\phi_1(x,p) = x$ and $\phi_2(x,p) = (df)^\ast p$. So let us compute $d\phi$ in these coordinates:
$$
d\phi = \left(\begin{array}{rr} 
\frac{\partial \phi_1}{\partial x} & \frac{\partial \phi_1}{\partial p} \\
\frac{\partial \phi_2}{\partial x} & \frac{\partial \phi_2}{\partial p} 
\end{array} \right)
 = \left( \begin{array}{rr} 
\mathbb{1} &  0 \\
\frac{\partial \phi_2}{\partial x} & (df)^\ast
\end{array} \right).
$$
Then basic linear algebra shows that $(df)^\ast$ injective $\implies$ $d\phi$ injective, and $(df)^\ast$ surjective $\implies d\phi$ surjective. Equivalently, $df$ surjective $\implies d\phi$ injective, and $df$ injective $\implies d\phi$ surjective. The answers to your questions (1)-(4) should be more or less obvious now, so I leave the details to you.
