Lived in Lincoln for 10 years (until about 10 years ago) and still get there 3 or 4 times a year. I'd second most of what Dessiato says.
The 'tourist trail' has Steep Hill (does what it says) with a variety of indepedent sort of shops and so on, there's the Cathedral and Castle, Lincolnshire Life Museum just a bit further out (not open Sundays these days) - 'The Collection' is just off the main tourist trail. The former mental hospital is now 'The Lawn' complex - on Union Road opposite the Castle - although last time I went, most of the shops etc had shut.
For pubs, I'd suggest the
Victoria on Union Road, close to Castle / The Lawn. The ones by the River Witham (presume that's what JHE means) in the city centre were a bit touristy.
Slightly further off the trail but worth a visit is the
Dog & Bone (formerly the "gay dog") off Monks Road just to the east of the City centre - purveyors of Lincolnshire's Batemans beer. It seems to have changed hands since I last went, so don't know if they still do Salem Porter straight from a barrel in the cellar...
There's also a
transport museum on the southern edge of the city - do-able by bus, and with a special event on Sunday 2 November which will involve lots of buses to and from the city centre.
Much more here on
Visit Lincolnshire
There's quite a bit more night-life (if that's your thing) than there was 10 years ago, what with the university growing.
You could probably do a very quick visit to the Castle and Cathedral and such in a day, but it will be a heck of a rush if you're coming all the way from Kent.
Newark is possibly also worth a few hours' mooch - market town with Castle (remains of - not quite been the same since the civil war) lots of antique shops and Gannets Cafe (35 Castle Gate) - purveyor of rather good cakes.