Can a first-order autonomous system not at a fixed point, transition to a fixed point? The following is from Introduction to Dynamics, by Percival and Richards:

At each zero $x_{k}$ of the velocity field $v\left[x\right],$
$$v\left[x_{k}\right]=0,$$
so that a system initially at $x_{k}$ remains there for all time. The points $x_{k}$ represent states of equilibrium: they are named fixed points. At all other points the state of the system changes. A system in an open interval between two fixed points cannot pass either of them. Such open intervals, together with those that extend from a fixed point to infinity, are invariant, as are the fixed points. Such fixed points and intervals represent invariant sets of states which are defined by the property that if any system is in such a set at some time, then it remains in that set for all times.

The discussion pertains to first-order autonomous systems in 1-dimensional phase space.
The mathematically correct reading of their statement appears to be that a system in some invariant set of states which is not a fixed point will not transition to a fixed point.  That is, the idealized system will never come completely to rest in a finite amount of time.  Is that correct?
 A: 
the idealized system will never come completely to rest in a finite amount of time. Is that correct?

Yes, a fixed point $x^*$ in such a system can only be reached asymptotically.
The simplest way to see why it's so is probably through the uniqueness of a $C^1$ differential equation: $dx/dt\equiv \dot x(x) = 0$ admits as solution the constant function $x(t)= x^*$, thus, by uniqueness, there cannot be a different solution; in particular, there can not be a solution that has $x(t_0) \ne x^*$ and $x(t^*)=x^*$ for some finite $t^*>t_0$. If an equilibrium point can be reached in finite time, then the system is not a smooth first order autonomous ODE.
Another way of proving this results is via the Hartman–Grobman theorem, which tells us that, for a smooth system, we can use its linearization to assess the system's behavior in the vicinity of a fixed point $x^*$ that's hyperbolic (i.e., for which $\dot v(x^*)\ne 0$). And, for a autonomous $1$-D system
$$ \dot x(x) = v(x) $$
the linearization $y\equiv x-x^*$ about the fixed point $x^*$ is
$$ \dot y = A y,$$
where $A\equiv \dot x(x^*)$, and whose solution is
$$ y(t) = y(0) e^{At}.$$
Which means that, for a stable $x^*$ (i.e., $A<0$), $y$ approaches $0$ (i.e., $x$ approaches $x^*$) only asymptotically.
