Hello Verna!
Unfortunately, the distance between Miyazaki Cruise Port and the Castle is quite far. Moreover, there is no available bus to take you to the castle. You need to use the train.
I'm assuming that you'd be in Miyazaki for one day (8am - 4pm). Here's what I think would work if you want to go to Obi castle.
1. From the Port, you would need to take a taxi to Miyazaki Station. (there is no bus available, except one in the morning at 8:46 AM. the trip would only take 10 minutes)
2. From Miyazaki Station, it would take approximately 1.5 hours to go to Obi Station (take the Nichinan Line towards Aburatsu).
3. When you reach Obi station, the castle itself is still about 1km away. If you want, you can ride a bus from the Obi Station to the Obi Castle Ruins.
If you choose to go to Aoshima Shrine from Obi Castle, the journey itself would be an extra 1hour 15minutes.
However, the journey from Miyazaki Station to Aoshima Shrine itself is reachable by bus, and would only take around 30-40 minutes.
Alternatively, I would suggest you take the train from Miyazaki Station to Aoshima station (take the train on the JR Nichinan Line, 30 minutes, 370 yen). From there, the shrine is 5 minutes walk from Aoshima Station.
Here's a bus map of Miyazaki, in case you're searching for more tourist destinations that are more suitable for your time frame.
Hope this helps!