How to estimate the number of microstate that a certain volume of gas go through in a period of time under certain temperature?

It is been explained in section (2.2.2) of Shcwable. After a bit of simple math, you get begin{equation} Omega(E)= frac{V^N2pi m(2pi mE)^{3N/2-1}}{h^{3N}N!(3N/2-1)!} end{equation}

for the number of eigenstates on a given energy shell. Then you can calculate $Delta Omega$ by relating the velocity of the particles at the given temperature to it.

and notice that for driving the above equation we start from: begin{equation} Omega(E)=frac{1}{h^{3N}N!}int_{vert H(textbf{x},textbf{p} )-E vert<varepsilon}d textbf{x}dtextbf{p} end{equation}

which is the volume of the energy hypersurface (the same as the number of microstates) where $textbf{x}=[x_1,x_2,...]$ and $textbf{p}=[p_1,p_2,...]$ are the positions and momenta of the particles. As we see in the above equation, considering that $Omega(E)$ does not depend on time or any other function other than system Hamiltonian, we can apply it to the close systems with the conserved energy, or microcanonical ensemble.

If you are still interested in finding the time-dependent number of microstates for a canonical system with the Hamiltonian $H_S$, embedded in a bath with the Hamiltonian $H_B$ and interaction $H_I$, you should change the boundary of the above integration to the $vert H_S(textbf{x},textbf{p} )+H_B+H_I-E(t) vert<varepsilon$, which I believe there is no direct solution for it.