I visited Karatsu castle when I was in town for Karatsu Kunchi festival. I don’t remember for what reason (maybe because it was too early in the morning), the castle itself was closed, so I didn’t get inside. But I got to enjoy the views of the beautiful structure itself, the bay and the city – the castle stands on a hill, so the views are good even if you’re at the foot of the castle and not on the top floor.
It’s about 10 minute walk from Karatsu station. You can climb the hill on foot (which I did on my way there) or use an elevator for 100 yen (which I did on my way back). Yes, the elevator goes up and down the hill, it’s not inside the castle.