York Minster and Castle museum

With only one full day you'll have to move quickly to do justice to York, it really deserves more time than one full day but if that's all you have then you won't want to give the Castle Museum any more than 2 hours, in fact, guidebooks like Michelin don't tout the Castle Museum as much as Rick does and you may find you give it a bit less than his 2 hours, which will give you more time for the Minster. The main attraction is, of course, the Minster. And allow time for the city wall walk. Those 3 things will take all of your day. Given your schedule I'm not sure the combo pass makes sense for you.

My daughter lived in York for two years, so I feel qualified to say that the main reason for going to York is the Minster. On all my trips to see her, I easily spent most of one day there. It is awe-inspiring. And don't miss the section whereby you go under the minster and see the Roman ruins. To me, the castle museum was not all that interesting, but a walk along the city walls is very nice, too.

The Minster and walking the wall, if the weather is good can take most of the day. There's a great walking tour of the town in the late afternoon. If you had an extra day, Castle Howard is fun, easy bus ride from York.

I agree with everyone else. The Minster is definitely worth the most time. I really enjoyed the Castle Museum, but we had 4 days. It is located quite a walk away from the Minster. However, a good piece of the wall (between Monk Bar and Bootham Bar) is near the Minster, as is the Shambles. The Yorkshire Museum is also much closer. If you are there at the right time, you can also take one of the free walks Rick talks about. I think you'd get more out of York piecing together some of those than walking across town to the Castle Museum.