How can I know when I have added an excess of magnesium oxide?
$\ce{MgO}$ is insoluble in water (solubility: 0.0086 g/100 mL). On the other hand, $\ce{MgSO4}$ is soluble in water (solubility: 25.5 g/100 mL).

The reaction that is taking place is an acid-base reaction:

$$\ce{MgO + H2SO4 -&gt; MgSO4 + H2O}$$

Now for your question, how would you know when excess $\ce{MgO}$ is present? Here you are trying to make $\ce{H2SO4}$ the limiting reagent. When all the $\ce{H2SO4}$ has reacted, no more $\ce{MgO}$ can further react. Therefore, you will see solid in your reaction mixture that cannot be further dissolved.

At the end of your reaction,

$$\ce{MgO_{excess} + H2SO4 -&gt; MgSO4 + H2O + MgO_{remaining}}$$