You're in luck here, it all looks fairly simple.
The Dormy inn is a 9 minute walk to and from the station. Here is a Google map showing the route that you can walk.
There are 2 or 3 buses every hour going from the station (line number 5) to the port and 1-2 every hour coming back from port to station. The bus is not a JR bus, and will cost ¥200 one way. It's a short ride though, taking approx. 16 minutes. As your hotel is so close to the station, your best bet is to head to the station from your hotel and get the bus there.
Here is a translated bus timetable for you from Niigata station to Niigata port (bear with it, google translate is pretty rubbish with Japanese). The table on the left is the route from the station to the Sado kisen niigata ferry port (called 'Sado Kisen wear' in the translation), and the table on the right is the return from the port (again, sado kisen wear) to the JR train station.
This timetable is until the end of march, so it's worth checking back at a later date to make sure the times are still correct (If anything there will be more buses after march due to the start of the tourist season).
Finally, here's some more general access information for the port itself.
Hope this helps!