Would building be capitalized when used in a sentence as Empire State Building?

In this case the whole phrase Empire Satte Building is a proper noun. No one mentions,"have you been to the Empire State in N.Y.C.?". Therefore Building is part of the whole proper noun and must be capitalized as a proper noun should be.

Yes, "building" would typically be capitalized when used as part of a proper noun like "Empire State Building." Capitalization rules generally dictate that proper nouns, including names of specific buildings, should be capitalized. However, it's important to note that this rule may vary depending on the style guide or specific context in which you are writing. When in doubt, it's always a good idea to consult the specific guidelines of the style manual you are following (such as MLA, APA, or Chicago Manual of Style) or the preferences of your instructor or publisher.