If it's going to shrink, I would rather have it do it BEFORE I use it in my quilt. So I would wash it and put it in the dryer.
Is it regular width (44"?) If so, you will need to sew two pieces together. Since you're not sure how much it will shrink, I'd probably cut about 3-1/4 yards. Before allowing for shrinkage, you will end up with a backing of about 58" x 84". If you think it might shrink a lot, AND you're having a longarmer quilt it (we like 6 - 8" larger for the backing), then I would cut a 3-1/2 yard piece.