Here are some of the popular places where ALTs like to stay in Nagasaki.
Casa Noda
This hostel has a very eclectic feel to the lobby decor. There is even a hammock! If you want breakfast, they have jam and toast available in the morning. You can stay in the larger dorm rooms, typically eight people to a room, or a private room if you want to pay extra. The beds are fairly comfortable with curtains for privacy. Be aware that it can get loud depending on who your roommates are, or when the staff comes to clean in the morning. This is the closest hostel to Nagasaki station, only about a 5-minute walk away.
http://www.booking.com/Share-gsTv0p
Casa Blanca
This hostel is decorated with a classy, modern color scheme and decor; it is probably one of the most Instagrammable places to stay in the whole city. The beds are comfortable, though there is very little privacy in some of the dorm rooms. It is located near Dejima and Chinatown, which makes it convenient for sightseeing, but it is 1.3 kilometers to the station, so you have to either take a short tram ride or walk for a while.
http://www.booking.com/Share-ZTwGyk
Hostel Akari
Akari is located slightly farther from the station than Casa Noda or Casa Blanca, but still close to public transportation and sights. It is a quick walk over from Megane-bashi, one of the most famous spots in the city. The rooms are cozy and comfortable, and the staff is friendly and knowledgeable about the city. There is a common area, including a library.
http://www.booking.com/Share-0YvVgN
First Cabin Nagasaki
If you want your own room but don’t want to pay full hotel prices, you can try First Cabin, a capsule hotel. This capsule hotel is particularly clean and new, with good service and even private TVs in the pods. It is located near Chinatown, a 15-minute tram ride from the station. The secret bonus of First Cabin is the onsen!
http://www.booking.com/Share-9RDesUJ
Dormy Inn
Dormy is a nice budget hotel chain. This hotel can be very affordable if you split the cost with a roommate. Be careful about the parking garage though, as it can get extremely expensive if you accidentally use more than your allotted time. It is located near Chinatown and the shopping streets, about 15 minutes from the station by tram.
http://www.booking.com/Share-sSkNQ3D
HafH
HafH is a company that offers lodging and open-work space for travelers. The primary locations in Nagasaki City are “HafH Nagasaki Garden” and “HafH Nagasaki SAI” (located in the Hamanomachi area). They also host various events open to foreigners and members of the community including coffee-making classes, game nights, workshops, seminars, etc.
https://www.hafh.com/en/properties/1
https://www.hafh.com/properties/2
Other places
There are many other hotels and hostels in Nagasaki! Your sempais can probably offer additional ideas. If none of the recommended places are available or fit your budget, try one of these travel sites.