Airport to 747 bus
This page should help you navigate in the airport to get to the 747 bus which will take you into the city.
Exit the plane and make your way to immigration. Avoid going to "connections". You'll see a US flag for the right line to get in. Show your passport and the form you filled out on the plane. After immigration, you go downstairs from there to get your baggage. Take your bags and the form and turn it in at the exit from the baggage area. Eventually, you'll pass through doors on your right into an area that looks like this:
All those people probably won't clap for you.
As soon as you go through the ropes, you'll see the kiosk that sells Metro tickets a bit to your left:
Up close it looks like this:
You'll need to figure out what kind of ticket to get. You can get a 1-day, 3-day, 1-trip, 2-trips, 10-trips, etc. More info is at the Montreal Metro website. The tickets are good for the 747 bus you'll take into town, and then on the Metro to get to the hotel.
From the ticket kiosk, go to the right towards this area:
You can see a sign for the 747 bus overhead.
To go to Hotel Dauphin, take the 747 to René-Lévesque / Jeanne-Mance. You'll need to walk a few blocks.
Or get off at the first bus stop (Lionel-Groulx / Saint-Jacques), and get on the Metro (same ticket unless you bought a 1-trip ticket). Take the orange line toward Montmorency. Five stops: get off at Place-d'Arms and walk to the hotel.
Or watch this nifty video: