Understand the processes involved

The first ionization potential of sodium is the energy required for: $Na(g) \rightarrow Na^+(g) + e^-$

Given: $\Delta H = +5.1\ \text{eV}$

Write the reverse process

Electron gain enthalpy of $Na^+$ corresponds to adding an electron to $Na^+$: $Na^+(g) + e^- \rightarrow Na(g)$

This is exactly the reverse of the ionization process.

Use the relation between forward and reverse processes

If a process requires $+5.1\,\text{eV}$, then the reverse process releases the same amount of energy: $\Delta H = -5.1\ \text{eV}$

Identify the correct option

Therefore, the electron gain enthalpy of $Na^+$ is: $-5.1\ \text{eV}$

So the correct option is A.

Comparison with stored answer

Stored correct answer: A

My derived answer: A

They match.