2-Bromo-1-chloro-4-nitrobenzene reacts with sodium ethoxide
The reaction over here is a $\text{S}_{\text{N}}(\text{Ar})$ reaction. Normally, nucleophilic substitutions do not occur, but in presence of strong nucleophiles and electron withdrawing groups, it is possible.

Now, coming to your reaction, let's see what really happens:

Consider the resonation structures of 4-chloro-3-bromonitrobenzene:

You can see that the resonance structures produce a carbocation at the chlorine attached carbon, but not at the bromine attached one. Nucleophiles always tend to attack electrophilic centers, so it is the chlorine that will get replaced.

Continuing with the reaction, we will get:


So, the product would be 3-bromo-4-methoxynitrobenzene.