Learn English – Why is New York City also called “the Big Apple”

In your second example, "city" should not be capitalized. Words for governmental or administrative units are only capitalized when they are used as part of a proper noun, such as the formal name of a city.

Your first example is correct so long as you're referring to the City of New York, as the formal name for New York. However, if I were simply using the word "city" to disambiguate and not as part of a formal name, I wouldn't capitalize it:

We're only going to the city of New York, not the rest of the state.

From grammatical considerations, some thumb rules for convenience could be applied. However, the question is largely one of style and less about grammar.

Firstly, when the title begins with an article, it is not always necessary to cite the article of the title. 'The New York Times' is referred to as the 'New York Times' in practice.

I went to the The New York Times office this morning.

Secondly, when the title beginning article is the same as the article required before the title, it need not be repeated.

I went to the The New York Times office this morning.

Thumb rule (2) provides an alternative to (1) so you may chose between capitalizing the article and not.

In the case of different articles as in

I am reading a The New York Times article.

use thumb rule (1). Here the grammatical requirement of an article is more important.