Regulations on a stay exceeding 90 days fall within the competence of national authorities, therefore there is no uniform procedure among the Schengen Member States. Foreign nationals requesting entry to Hungary with a purpose of a long stay may submit a residence permit application at a Hungarian Consulate operating in the country of residence. The General-Directorate for Aliens Policing of Hungary makes a decision in residence permit applications.
In the case of a successful application, the applicant will receive a visa valid for a single entry and a stay of 30 days in Hungary, where the request for the issuance of the residence permit could be lodged at the local unit of the General-Directorate for Aliens Policing.
A valid residence permit from one of the Schengen countries can be used as a short stay visa within the Schengen Area.
Family members of EU/EEA nationals with the right of free movement may enter Hungary with a short-stay uniform visa (C), even if they intend to stay in Hungary for more than 90 days. In such cases, within 93 days from the date of entry, the family member shall apply for a residence card at the regional office of the Directorate-General for Aliens Policing.
Third-country nationals who are exempted from visa requirements (listed in Annex II of (EU) 2018/1806 Regulation) might lodge their applications for a residence permit in Hungary, directly at the regional Directorate for Aliens Policing in Hungary.
Please check the requirements for the residence permit application at the webpage of the Directorate-General for Aliens Policing:
http://oif.gov.hu/index.php?lang=en