Yes, you will notice the weight. What you do about it will depend on your riding style!
For most riders who have lived their whole life on a two stroke, the four stroke will feel yukky when they first ride it. It'll feel top heavy, sluggish and pig-like.
It's really not that bad, it just requires a different riding style and mind-set to make the bike into something great.
If you try to ride it like a two stroke, you'll never like it. If you ride it like what it was designed to be, a great trail thumper, then you'll have a good time!
BTW, to really bring out the best in the DRZ, get a Baja Designs kit and make it street legal! These bikes love to explore two track.
You are getting the Electric Start model, aren't you? You'll like it MUCH better with that happy button.