What time do you mean by early dinner? On a Sunday you're not likley to find a busy top end restaurant in the evening. The most obvious that are open being The Stockbridge Restaurant, Wedgewood and The Number One at the Balmoral. If you're just looking for a laid back affair then I'd head out for a Pub lunch on the Sunday. We had a great lunch at the Sheeps Heid a coupld of Sunday's ago. It's not right in the city centre so you'll probably want to get a taxi there and you'll need to book as it can get very popular
best chance of mingling would be in a pub, though the locals may be staring spaced out at the football!
stockbridge doesn't get many tourists, bailies is an old fashioned pub, hectors is more of a modern pub/bar
Stockbridge tap opposite hectors is a friendly pub, though i don't think they sell food these days. If you're on a cruise will you be in Leith? Plenty of pubs around there - i'm not so familiar with the area, king's wark is a pub known for its food, the malt shovel next door used to at least serve stoves
Leith and Stockbridge are a bit like communities in themselves - village like has been the term applied to stockbridge - the locals are different in both and from pub to pub! - so you might need a tour of Edinburgh to get representative sample of what the locals are like.