Currently I hear a lot of the expression heritage structure.
Though dictionaries generally define a monument as a commemorative, the term is also used in the cultural context for a structure of heritage value:
"• Monumental heritage of exceptional cultural value;" (UNESCO: Historic Urban Landscape)
Wikipedia includes the 'heritage' sense of usage:
... which has become important to a social group as a part of their remembrance of historic times or cultural heritage, or simply as an example of historic architecture.
monument a building, structure, or site that is of historical importance or interest
I feel the example sentence sounds better with a noun instead.
The government building is a heritage monument. Many people flock to visit it every year.