I'll be honest, I'm more of a football girl than a rugby girl, but places we go to that show big screen sports include Frames, Basement Bar and The Globe. Brennan's Bar often have sports on too. Hopefully a rugby fan can advise you further though!
As for restaurants, define "affordable": for some people it's a tenner, for some people it's £50. Also what kind of food do you like? We have everything from Lebanese to Thai to Indian to Irish Fusion (to name but a few). Without any further guidance some of my favourites are Northern Whig, Made In Belfast and Gingeroot. Other people often recommend Ginger Bistro, Rhubarb, Mourne Seafood Bar, No 27 Talbot Street, Hill Street Brasserie, Deanes, Cayenne, Nick's Warehouse... the list goes on!
Bars/pubs again depend on what you're after. If you like trendier bars the Cathedral Quarter would be best for you (e.g. Spaniard, Cloth Ear, Social 21, Muriel's, Black Box, The John Hewitt, etc) whereas if you like a more laid back scene (which admittedly comes with its share of student types) try the "Golden Mile" area, e.g. Lavery's, Robinson's, The Crown, Katy Daly's, Auntie Annie's, etc. I like this website for checking what's on when you're in town - use the "select specific dates" option in the What's On section:
http://www.belfastmusic.org/