= Solution
Adiabatic photon redshifting gives $T\propto a^{-1}$. Using the stipulated <temperatures>,
$$
\boxed{1+z_{\rm rec}\simeq1000,\qquad z_{\rm rec}\simeq999.}
$$
In the flat dust approximation of part <S2/3/c>, $t=t_0a^{3/2}$, so $t_{\rm rec}\simeq(2/3H_0)1000^{-3/2}\simeq2.75\times10^5$ years. This approximation omits the significant radiation contribution near the actual <recombination> epoch.
The elementary travel-distance estimate is $ct_{\rm rec}\simeq8.4\times10^4$ pc, whose corresponding comoving scale today is about $84$ Mpc. In an expanding universe, however, that is not the exact proper particle-horizon <radius>. Integrating the <null geodesic> gives
$$
d_{\rm hor}(t)=a(t)c\int_0^t\frac{dt'}{a(t')}=3ct
$$
for dust. Thus the proper horizon at this epoch is about $0.25$ Mpc and its present comoving size about $250$ Mpc. The factor three accounts for expansion during light propagation. These scales describe the limited causal domain in the simple decelerating model. The near-uniform CMB <temperature> across much more widely separated last-scattering regions motivates the <horizon problem>; the question's rough $ct$ estimate and the exact dust horizon should not be conflated.
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