A method of fieldwork where someone studies a culture by doing what the people of that culture do is called:

the choices are

interviewing

archival research

participant observation

surveying

Participant Observation

The very definition of Participant Observation is a method for data collection in research where the researcher introduces themselves into the field of study in order to collect more detailed and meaningful data for a prolonged time period (months to years). All of the other methods mentioned don't require being a participant in the culture being investigated.

Therefore, your best answer is participant observation.