Agree with the others regarding the Titanic anniversary (except that it was in 1912, not 1911 - 100 years exactly, which is why it's supposed to be quite a big deal). If it stays on schedule, the new Titanic Belfast visitor experience should be newly opened during your stay, so that could be worth a visit:
http://www.titanicbelfast.com/
For anything else happening in the city, check out the Belfast City Council's Events page. There's not much on there now for your dates but more will be added closer to the time:
www.belfastcity.gov.uk/events/index.asp
As for things being opened or closed, it's kind of hard to tell right now. Belfast has restricted trading hours on Sundays anyway (1pm-6pm only) so apart from some cafés or restaurants, and places like museums, not a lot is open on Sunday mornings. Add in a public holiday like Easter and it can go one of two ways: either more places close to allow their staff time off, or more places open to take advantage of an influx of visitors. I would hope that with the Titanic anniversary going on at the same time, it would more likely be the latter, but we probably won't know for sure until a few weeks before I'm afraid.
Personally I think that the tour to the Causeway is a great idea for a Sunday - just make sure they're definitely running them that day. I did it myself recently with some family who visited (you can see my trip report at the link below) and it was a great way to occupy ourselves whilst everything in the city was closed:
tripadvisor.com/ShowTopic-g186470-i491-k4985…
A couple of weeks after that, we had some more visitors come to stay in Belfast for a flying visit and we ended up visiting the Ulster Museum and Botanic Gardens instead on the Sunday morning, since those were some of the only real options available to us without a car. My favourite of the two was definitely the Causeway bus tour.
As for places to eat, there is a good little list of places that are open on Sunday mornings on this pdf (see page 3):
gotobelfast.com/PDF/Summer_Sundays_2011.pdf
Once trading hours start at 1pm you should be able to find somewhere to eat no problem. Of course if you do go for the causeway tour, you'll really only need to worry about breakfast and dinner anyway, since you'll be able to get lunch on the tour.
If you'd like any suggestions for dinner, give us an area of the city, an idea of price and a type of cuisine and we'll happily chip in with some ideas for you.