There is no direct bus to Norrköping Airport but there are a few that passes by.
From Söder Tull
Bus 116 bus stop Smedbygatan (1km from Norrköping Airport)
Bus 440/441/444 bus stop Flygplatsen (200m from Norrköping Airport)
Passengers from outside Norrköping
Bus traffic in Norrköping is run by Östgötatrafiken. You can travel all the way from Tranås/Mjölby/Linköping to a bus stop close to Norrköping Airport on one ticket.
From Norrköping airport to the city
You have to walk to one of the bus stops nearby and from there take the bus in to the city center.