The bars and clubs are all pretty much closing at 3am, as by Quebec law they are not allowed to serve alcohol after that time. Several fast food restaurants and coffee shops will have locations that stay open 24hrs. You will probably find a McDonald's or a Tim Horton's in the downtown area for that.
The Casino of Montreal is open 24hrs a day, so that's one place if you are over 18. You could bring your luggage with you and ask if the casino can store them in the coat check area.
There's also a cafe called "Ciné Express" on St. Catherine Street near Rue Saint Marc that has coffee, pastries, and a booths in the back where you can watch movies. No, nothing shady. It's supposed to be more for the artsy crowd or the insomniac. I hear on Tuesdays they used to have a video gaming tournament into and through the night.
By the way, the bus station will drop you off around the Berri-UQAM area, and at 3am the main city buses and the subway will be stopped. You will only have access to taxis and night bus service. If you walk west two streets, you will reach St. Denis. If anything is open at 3am, it would likely be a place along there.
This Montreal public transit website may help you for maps:
http://www.stm.info/English/info/a-plans.htm
This is the map specifically of where the bus will let you off.
http://www.stm.info/English/metro/Berri-UQAM.pdf