Federal Prison Oversight Act This bill establishes an inspections regime for the Bureau of Prisons (BOP). The Department of Justice (DOJ) Office of the Inspector General must conduct risk-based evaluations of BOP facilities. An inspection of a facility may be announced or unannounced. Higher risk facilities must receive more frequent inspections. The bill (1) provides for the establishment in DOJ of an Ombudsman who may receive complaints, make inquiries, and recommend actions, and decline to investigate or take action, as specified; and (2) prohibits BOP retaliation against any person or entity that has instituted or caused to be instituted any proceeding, investigation, or inspection under this bill. The ombudsman shall coordinate with the BOP to educate incarcerated people, their representatives, and the public about the existence and functions of the ombudsman.