Bob has the good advice. Unless they've added personnel (doubtful) the gates are unmanned that early in the morning (same thing in the evening). "The early bird gets the worm." Just have plenty of coffee to wake yourselves up! :D
As far as being too early to drive the entire Sun road, although that is usually the highlight for most folks there are still plenty of nice vistas and hikes. You're likely to see a ton of critters besides. Take warm clothing, though.
And remember, just about all of that part of Montana is beautiful. If you like art museums, visit the C.M.Russell Museum in Great Falls.