1. Given reaction

$\text{FeO}(s) + C(\text{graphite}) \to \text{Fe}(s) + \text{CO}(g)$

For spontaneity,

$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ < 0$

The minimum temperature for spontaneity is when

$\Delta G^\circ = 0 \Rightarrow T = \frac{\Delta H^\circ}{\Delta S^\circ}$

So first compute $\Delta H^\circ$ and $\Delta S^\circ$ for the reaction.

2. Calculate standard enthalpy change

Using

$\Delta H^\circ_{\text{rxn}} = \sum \Delta H^\circ_f(\text{products}) - \sum \Delta H^\circ_f(\text{reactants})$

Products:

$\Delta H^\circ_f(\text{Fe}) = 0$ $\Delta H^\circ_f(\text{CO}) = -110.5\ \text{kJ mol}^{-1}$

Reactants:

$\Delta H^\circ_f(\text{FeO}) = -266.3\ \text{kJ mol}^{-1}$ $\Delta H^\circ_f(C) = 0$

Therefore,

$\Delta H^\circ_{\text{rxn}} = [0 + (-110.5)] - [(-266.3) + 0]$

$\Delta H^\circ_{\text{rxn}} = -110.5 + 266.3 = 155.8\ \text{kJ mol}^{-1}$

3. Calculate standard entropy change

Using

$\Delta S^\circ_{\text{rxn}} = \sum S^\circ(\text{products}) - \sum S^\circ(\text{reactants})$

Products:

$S^\circ(\text{Fe}) = 27.28\ \text{J mol}^{-1}\text{K}^{-1}$ $S^\circ(\text{CO}) = 197.6\ \text{J mol}^{-1}\text{K}^{-1}$

Reactants:

$S^\circ(\text{FeO}) = 57.49\ \text{J mol}^{-1}\text{K}^{-1}$ $S^\circ(C) = 5.74\ \text{J mol}^{-1}\text{K}^{-1}$

Thus,

$\Delta S^\circ_{\text{rxn}} = (27.28 + 197.6) - (57.49 + 5.74)$

$\Delta S^\circ_{\text{rxn}} = 224.88 - 63.23 = 161.65\ \text{J mol}^{-1}\text{K}^{-1}$

4. Find threshold temperature

Convert enthalpy into J mol$^{-1}$:

$\Delta H^\circ = 155.8\times 10^3 = 155800\ \text{J mol}^{-1}$

Now,

$T = \frac{\Delta H^\circ}{\Delta S^\circ} = \frac{155800}{161.65}$

$T \approx 964.74\ \text{K}$

For the reaction to become spontaneous,

$T > 964.74\ \text{K}$

Hence the minimum temperature is approximately

$\boxed{965\ \text{K}}$

If integer truncation is expected, one may write $964$, but physically the smallest whole-number temperature at which spontaneity starts is $965\,$K.

5. Comparison with stored answer

Stored correct answer = $964$

My calculated threshold is $964.74\,$K, so the least integer temperature making $\Delta G^\circ < 0$ is $965\,$K. Therefore I do not agree with the stored answer.