Another vote for Columbus Schwinns! I have a 1989 Circuit I picked up last year that was complete except for the little black rubber nub that goes in the rear brake pivot bolt. I got a longer stem and seatpost and a bigger handlebar, but otherwise it's original:
That one set me back $450, but it had everything original!!
Another option would be a Centurion. I got this 1985 Ironman for $300, in ready to ride condition, though I did drop some coin recabling it and such:
While we're on Japanese bikes, there's also Lotus. These used high quality Tange steel, brazed by Tsunoda. Here's a 1982 Lotus Classique that had spent most of its life indoors. It was $175, though I did need to get a longer seapost and a saddle that wasn't a torture device.
So, really, you can probably leave most of the $1000 in the bank, if you know what you want, and look carefully. If you're not mechanically skilled, you might want to look for a bike that is already fixed up. Rebuilding old bikes is easy and fun, but it's not for everyone.