Hi.
If that was me, I'd probably drive to ARN earlier and get a room at the airport. There are several hotels at or near ARN, and also a cheap (more or less) place inside the Sky City building (between Terminals 4 & 5) called Rest & Fly. The rooms are more like cabins without a window, but perfectly adequate for a few hours sleep. I used it a while back, had no issues. There's a link to their website under the "Hotels" section at arlanda.se

ARN, most of the terminals anway, is open 24 hrs, but I wouldn't count on getting much real rest in the public areas. If you don't want to be completely exhausted when you get to your destination, I wouldn't hang around there all night.

As for parking, I always use the "official" long term parking near Radisson Arlandia and the rental car lots, but I know there are a little cheaper options slightly further from the airport. Don't know their locations, but a google search would probably find something.