Most definitely visit Old Montreal. It's beautiful, and there's lots to do.
There's so many things to see and do in Montreal. They've got a cool planetarium. I'd check online to see if any festivals will be going on while you're there.
The nightlife is great, as is the shopping and food. Spend time on Ste. Catherine Street. There's always lots going on there. Try Montreal bagels, smoked meat and poutine (don't get that from a fast food chain, get it from a real Montreal place). Go to Dunne's for their great cheesecake.
There's lots of nice architecture, ie Notre Dame. Mont Royal is a nice site to walk around.
If you like amusement parks, there's always La Ronde.
They've got a good symphony and plenty of theatre, too. You might want to check Place des Arts to see if they've got anything good playing.
I loved walking up Mont Royal and the view at the top is lovely. Take the metro to get around (& see the artwork). I don't know if you can still take a tour of the Olympic sites there (from 1976), but we enjoyed doing that. The Botanical Gardens are very nice, and there's a very impressive Bonsai collection there. We also stopped at the St. Lawrence Seaway canal locks, as we like watching the ships come through.
You're going during a much nicer time of year than when I went. We went up there for a Rolling Stones concert at the Olympic Stadium in early December several years ago. It was SSOOOOO COLD. Our seats were up top, too, of course. None of us were very interested in venturing outside late night because of the weather.