I think I would hand applique over the holes -- maybe bees and a beehive. You might be able to find a fabric that has bees and hives that you could cut out to use. Although it would be done by hand, I think this would be a lot less intimidating than taking the quilt apart to replace the sashing. It would also leave the quilt in a more original state. Because the fabric is quite old, I would also either line it with a thin washed muslin or cover it with tulle as QuiltnNan suggested. Either of these (or both) would extend the life of the quilt. I would probably use Quilter's Dream Request cotton for the batting to keep it thin and light in weight (not too puffy).
Edit: Appliqued flowers would work too to cover up the holes.