You will always get some competition between E2 and SN2 reactions, if both reactions are possible. By using a really good nucleophile, you can get almost complete SN2 reaction, and by using a strong base with no good nucleophile, you can get almost complete E2 reaction.
The question you were given really represents the two extremes of steric hindrance and their effect on the reaction. Alkoxide ions in general are good nucleophiles, and bromide attached to an alkane is a good leaving group. Since your alkyl bromide is the same in both reactions, and both alkoxides are approximately the same strength as bases, the elimination reaction will take place at about the same rate under both reaction conditions. What you are looking at is how fast the SN2 reactions will occur.
If the alkoxide is not sterically hindered, the SN2 reaction will be quite a bit faster than the E2 reaction. The carbon-bromine bond is strongly polarized, and attracts the negative charge of the alkoxide better than the proton on the next carbon. If the alkoxide is sterically hindered, however, it is too big to get close enough to the carbon, and has to be content with interacting with a proton instead. In general, the reaction of ethoxide or methoxide will be a rapid SN2 substition, while the reaction with t-butoxide will be a slower E2 elimination.
It is with the intermediate compounds that things like solvent effects become important. If you had a secondary alkyl bromide rather than a primary one, or a secondary alcohol rather than a primary one, or even something like t-amyl alcohol where it is primary but has a large group on the next carbon, you may slow the SN2 reaction down to the point where you are getting a mixture of substitution and elimination products. In that case, you may need an even stronger nucleophile to drive the reaction to the substitution product. Although ethoxide is a strong nucleophile, as you say, in an ethanol solution there is still some ion-solvent interactions that need to be overcome for the reaction to occur. In this case, you can make the nucleophile even stronger by using a solvent which does not have such strong solvent-ion interactions. This is a very small steric effect, however, compared to an actual covalent bonded group on the alkoxide itself.