Pre-Theatre near Lyceum on Friday
I think it's difficult to find 'classy' and not tooo expensive. There's a branch of Cité near the Strand. It's a chain but the food is not bad and I've seen famous people eating there, if that means anything. That said, I'm not sure it meets your 'classy' requirement. There are places on nearby Great Queen St. There is Great Queen Street, and you can check their website for prices and menu.