Final answer:
Santa Fe is the city that has been a U.S. capital for the longest time and houses the Palace of the Governors, the oldest public building in the U.S.
Step-by-step explanation:
The city Santa Fe is the answer to the student's question. Santa Fe has been the capital of New Mexico since it was claimed as a Spanish colony and is home to the oldest public building in the United States, the Palace of the Governors. This historic building was constructed in 1610, making Santa Fe the correct choice as both a former U.S. capital and for having the oldest public building.
In contrast, cities like Boston, Philadelphia, and New York City have also played important roles in the history of the United States, with Philadelphia serving as a capital for a time and hosting significant events such as the drafting of the Declaration of Independence. However, none of these cities houses the oldest public building. Therefore, we can confidently select Santa Fe as the city that fulfills both criteria mentioned in the question.