I don't know which apartment to choose. The price/rate/cost in this city is just insane.
What the most appropriate option?
Generally, without further context I would understand the terms this way:
Price is to purchase the apartment outright. e.g. "The prices for apartments in this city are insane."
Rate is paid every week/fortnight/month for the rent. e.g. "The rental rates in this city are insane."
Cost might be understood to refer to general cost of living expenses, e.g. "The costs in this city are insane."
So, Rate - but I would use the more specific term rent if that's what you're referring to. e.g. "The rents in this city are insane."
On the other hand, in a city where the majority of apartments are very expensive and generally rented, and if you appear to be a student of low means, I would understand your "prices" to mean rents, but this is making an assumption.