The only cars I can think of that might still have some life left in them for under $1000 would be...
Older full-sized 3.8L V/6 GM cars like Buick Lesabre, Olds 88, or Pontiac Bonneville.
Older mid-sized 3.8L V/6 GM cars like Buick Regal, Buick Century, Olds Cutlass, or Pontiac Grand Prix.
Older full-sized V/8 Ford/Mercury cars (that were not former police cars) like Crown Victoria or Grand Marquis.
Note: Many of the above mentioned car models were owned by older people who took good care of them and kept them in the garage for much of their life.
Older Ford Explorers or V/6 Ford Rangers.
Forget trying to find a Toyota or Honda with any life left in it for under $1000 unless you are willing to spend another $1000 (or more) on it to make it minimally roadworthy.