I'd start at the Cafe Procope (091-232 3848), which is very friendly. I'd do the crossword and have a double espresso. There's a little pub next door called The Crown Posada (091-232 1269). It's almost completely unspoilt with fantastic stained glass windows, and no music. I'd have a bloody Mary.
I'd then have an early lunch at the best restaurant in Newcastle, 21 Queen's St (091-222 0755). It's very similar to a restaurant in Belfast called Roscoff's (0232 331532) which I'd recommend to anybody. Very cool, very white and light.
To really get a feel of the beauty you have to go to Hadrian's Wall which I've always found fantastically moving. It's so bleak. If you stand on the wall and look north you feel what those poor Roman soldiers must have felt.
My other trip would be down south to Durham Cathedral for evensong. An hour's meditation in what must be the most beautifully- sited cathedral in Europe.
I'd return to Newcastle and go to the theatre, probably The Playhouse (091-230 5151). Afterwards there's a very reasonable, wonderful Thai restaurant called the River Kaai (091-438 2212). They have sunken seats, you sink down into holes. You probably can't get out. They have a very simple pudding, bananas in coconut milk, which I think after a day on Hadrian's Wall would probably be rather nice.
Simon Russell Beale plays Ariel in 'The Tempest' at the Barbican, London EC2 (071-638 8891)