What is the equilibrium price in this case?
I have quickly skimmed the chapter and there does not seem to be much about bargaining models so as the question says you have to solve this using supply and demand.
We know that an equilibrium price on the market will occur when supply intersect demand. Given that the supply is fixed at $24$ and $25$ apartments respectively it will be just straight vertical line in each case. Now when it comes to demand, if we assume each person wants only one single apartment the demand will be given in step-wise fashion, with two steps given by reservation prices ${\\\$}500$ and ${\\\$}200$ respectively. So this situation will look like shown on the tikz picture I made below (supply is blue, demand is red):

At point where supply is equal to $24$ the equilibrium price given by intersection of supply and demand ($S=D$) is clearly ${\\\$}500$. However, when supply is equal to $25$, supply and demand will intersect at a whole range of different prices $[200,500]$.
Consequently, in this case using this simple supply and demand model we cannot say anything more that price will be something between ${\\\$}200$ and ${\\\$}500$. There are more advanced ways how we could model this situation which might yield unique price instead of range (maybe some models with explicitly modeled bargaining process or some matching models), but here the author of the book gives you hints by asking how demand would look like etc., that you are supposed to use this solution concept (also the first chapter uses only simple supply and demand models so that is another hint).