Here is some information but your problem is going to be whether these sights will be open on a Sunday, especially over the holiday season, so you should check the websites first:

In addition to the c15th/c16th old town which is pretty in itself (see the Town Hall and Maison Supersaxo) there are four main attractions:

Eglise Notre Dame de Valère - a c13th church cum fortress on top of one of the hills

Chateau Torbillion - a ruined castle on Sion's other hill. This is a bit of a climb but the views of the Rhone valley and mountains are awesome

The Cantonal History Museum

The Cathedral of Notre Dame du Glarier

Sion old town is nice just to wonder around. There's plenty of small shops, some selling local Valais produce and a myriad of bars and cafés. However, my guess is that it could be pretty deserted on a Sunday in the middle of winter but if you fancy a quiet walk around an old town centre then you should be okay.

On your way you could also stop off and visit Chateau d'Aigle (wine museum), the old town of Saillon and the Roman remains/museum and Tour de la Batiaz at Martigny. They're interesting to look at even from the outside. If you're taking the train, all of these are within walking distance of the train station except Saillon which doesn't have a station.