Montreal Piere Trudeau airport is one building, all domestic and international flights depart and arrive here. It is very convenient to navigate.
On the ground floor of this building, exit 27 [if you are paying attention to the numbers printed at each], is where the airport - down town bus, 747 AEROPORT P.E.-TRUDEAU / CENTER-VILLE, stops. There is a ticket machine close by, where you can buy a day pass for $10, and pick up a 747 bus time table which has a map of metro from the information desk close to the exit. The day pass is valid on the airport-down town bus as well as on metro, and all other buses operated by stm. Most Montreal sights are accessible by public transit. [I bought my ticket on the bus, I paid $10, exact change in coins. But unfortunately, my ticket didn't scan at any of the metro station. I went to the stm customer care, Service a la Clinetele, at the Berri-Uqam station and got a new day pass issued for me.]
There is metro station a short walk distance from the 747 bus' first stop, Linonel-Groux. Taking the metro is faster than staying on the bus, I think, to down town, the Saint-Laurent station.
Now, since it was Jan 1, I had prepared a list of places that are open.
Notre - dame Bascillica - Roman Catholic church, Gothic Architecture, with a beautiful chapel in the back [110 Notre Dame street, metro: Place-D'Armes] Jan 1 schedule morning mass, open for visitors 12:30-4:00 PM.
Bonsecours Basin Quays of Old Port [Ville Marie, metro: Champ-de-Mars] Skating rink admission: $9.00 Skate rental, $6.50+tax, themed music everyday in winter starting at 6PM, rink maintenance at 2 PM and maybe other times during the day, Jan 1 schedule 10 AM - 10 PM.
St. Joseph Oratory - Roman Catholic church - its dome the third largest in the world, Renaissence Architecture, escalators or taking the steps as a pligrimage to the chapel, overlooks the city of Montreal's Mount Royal Park. [3800 Queen Mary Road, metro: Cote-Des-Neiges]
Universite-de-Montreal, opened in 1878 with less than a hundred students in theology, medicine and law. [metro: Universit-de-Montreal]
I utterly enjoyed visiting all the places above; the city was clean, quiet and ready to welcome its visitors on Jan 1st.
Map of Montreal metro, run by stm.