A rental car is not essential. You would take the Shinkansen to Isahaya Station (one stop before Nagasaki Station), get off at Isahaya, and catch the Shimabara Railway at Ishaya Station. Take the Shimabara Railway to Azuma Station (which is in Unzen City) and then you can catch a taxi to the particular hotel that you reserve. It's best to directly contact the hotel that you plan to stay at and ask them at what station you should get off the train and if they can pick you up at the station. I have always traveled to Unzen by car (it's a very pleasant drive), so I don't know if the hotels have a pick-up service, though I would imagine that some of them do. Shimabara Railway website: https://en.shimatetsu.co.jp/railway/