There are a number of ways you can go after studying ancient history. You can go into academia, and teach history or a related field at the university or high school level. Also, there is the realm of archaeology, which is very closely tied to ancient history. Depending on the area of history that your student is interested in, there are jobs in the US working for CRM (cultural resource management) firms, which basically do emergency excavations ahead of construction projects. There are other jobs related to industry in this way, many major building or development companies or public institutions employ people who are knowledgeable in this field to work with people and consult for archaeological problems when construction or expansion or repairs are taking place.

Similarly, there are positions within the government that deal with cultural resources in this manner. State archaeologists are employed to deal with these things. I know quite a few people who are or were at one time employed by the National Park Service, dealing with various aspects of that, working on cultural resources in the parks and dealing with people as well.

There is also the realm of museum studies. There are a number of museums which hire people who are knowledgeable about ancient history and ancient artifacts.

If your student's interest is ancient history outside the United States, it certainly doesn't preclude any of the jobs on this list, but museums and teaching may be some of the things that will be closest to ancient history outside the US.