"Country" is to "national" as "city" is to…

A city, viewed as a governmental and political entity, is called a municipality (see Merriam-Webster), with corresponding adjective municipal and adverb municipally. For example, we often speak of "municipal elections".

However, I must say that the sentence "elections are held municipally every two years" does not sound anywhere near as good to me as the sentence "elections are held nationally every four years"; I don't know why. So I'd suggest instead:

National elections are held every four years, and municipal elections every two years.

Alternatively, in your example, you can use the phrase at the city level:

Elections are held nationally every four years, and at the city level every two years.

"municipal" is the word you're looking for.

Elections are held nationally every four years, and municipally every two years.

• municipal (adj) "of or relating to the government of a city or town"

e.g. Both national and municipal elections are held every four years in this country.

In some English speaking countries, mainly in the US, the term "local elections" is used more often than "municipal elections". See ngram

My preference would be for "locally"?

Depending on the context urban could be used to, I guess. But seeing the other answers, it's probably not what you wanted ^^