With that kind of rent money, you can afford to be picky!
Outremont is very nice and posh. But the hippest part of town is the Plateau Mont-Royal. Lots of great places to eat and great scenery of the movable two-legged variety to watch. NDG is not bad but a bit far removed from metro lines, unless you're near Monkland Village which is pretty hip and mostly anglophone.
Get a cheap but solid bicycle, and teach yourself to bike around Montreal. Best way to get around. I park my car on the South Shore now when I go into the city for work, and ride the remaining 20 or so kilometers on my bike. Great fun, and faster than a car downtown, with no parking hassles.