Suppose you have the autonomous ordinary differential equation $dx(t)/dt = f(x(t))$ with $x: \mathbb{R} \to \mathbb{R}$ and the initial condition $x(0)=x_0$. Then, assuming some regularity conditions, you get as solution the flow $\Phi(x_0,t):=x(t)$. To give a trivial example: If $f(x)=x$, then $\Phi(x_0,t)=x_0 \exp(t)$.

Now, I'm not interested in the trajectories for a given initial condition, that is in $\Phi(x_0,t)$ with $x_0$ fixed and $t$ variable; but in the map $x_0 -> \Phi(x_0, t)$ for a fixed $t$ (say $t=1$).

Given the function $f$, you can easily (at least in principle, by solving the ODE) get the function $\Phi(\cdot, 1)$. There are a lot of theorems about existence and uniqueness of this problem and analytical and numerical algorithms.

But how can one get $f$ out of $\Phi(\cdot, 1)$? Is this a well posed problem? Are there any theorems?

This problem is closely related to "interpolating" the $n$-fold functional iterates of $g$ (with $g^{[0]} = \mathrm{Id}, g^{[1]} = g, g^{[2]} = g \circ g, g^{[n+m]} = g^{[n]} \circ g^{[m]}$ for $n,m \in \mathbb{N}$) from $n \in \mathbb{N}$ to real values. If such an interpolation succeeds, on can get the ODE out of the flow $\Phi(\cdot, 1)$ by determining $\Phi(\cdot, 1)^{[\alpha]}$ for small $\alpha >0$. I have done some calculation, that give results, but lack in rigor.

For noninteger iterates of functions, a classical reference is http://www.math-inst.hu/~p_erdos/1960-07.pdf.