Maybe it's my poor English to blame, but!
From my own experience, the Afternoon Tea at Betty's is a real treat, therefore, not to be missed. What exactly have you booked? A table at the cafeteria, or the AT?
York Museum is pretty boring. It has left us underwhelmed. The Castle Museum - very special by all standards, and not to be missed!
Clifford's Tower is interesting, especially if you know what has happened there, and care...
The ramparts walk on the walls - free, and adorable. Especially, the Northern part - best views of the York Minster.
A boat cruise on River Ouse - extremely nice.
IMHO, York offers best ( and maybe cheapest ) pub crawl in UK. If you appreciate a good pint in a nice pub, York is the city to visit.
Hope it helped.
Yuri.