How to define time in a time-dependent solution?

A good reference for such things is Wald's General Relativity. In that book, you'll find the following result:

Theorem. Let $(M,g)$ (a manifold with a metric) be globally hyperbolic. Then

1. There exists a global time function, that is a map $t:Mto mathbb{R}$ such that $-dt$ is future-directed and timelike;
2. Surfaces of constant $t$ are Cauchy surfaces, and they all have the same topology $Sigma$;
3. The topology of $M$ is $mathbb{R} times Sigma$.

This result essentially tells you that if your spacetime is globally hyperbolic, then you have a uniquely defined time evolution. This "global hyperbolicity" is required in several results in GR and intuitively it means, well, that you have a unique time evolution. More formally, we can define what follows:

Definition. Let $(M,g)$ be a time-orientable spacetime. A partial Cauchy surface is a hypersurface for which no two points are connected by a causal curve in $M$.

Intuitively a partial Cauchy surface is like a spacelike hypersurface, on which you can define some initial data for time evolution. Moreover,

Definition. If $Sigma$ is a partial Cauchy surface then its future domain of dependence $D^+(Sigma)$ is the set of points $p in M$ such that every past-inextendible causal curve through $p$ intersects $Sigma$. Similarly you can define the past domain of dependence $D^-(Sigma)$. Then you can also define the domain of dependence $D(Sigma)=D^-(Sigma) cup D^+(Sigma)$.

Suppose that you have initial data defined on $Sigma$. Then the domain of dependence $D(Sigma)$ is intuitively the set of points on which you can propagate this initial data (say with some hyperbolic PDE). Finally,

Definition. A Cauchy surface $Sigma$ is a partial Cauchy surface with $D(Sigma)=M$. A spacetime with a Cauchy surface is called globally hyperbolic.

So a Cauchy surface is nothing but some hypersurface on which you can define initial data which can be propagated everywhere in your spacetime. Then the above theorem tells you a bit more about how such a spacetime will look like.