How can I replace these old ceiling tiles?
This type of tile is composed of pressed fiber material. They have edges that engage each other in a tongue-and-groove arrangement. They're typically installed on 1x2 battens running under the continuous seams. 

If you can find replacements, you can probably cut individual tiles out with a fresh utility knife. You'd then install the new tiles with construction adhesive. You'd need to support them for few hours, perhaps with temporary padded stilts. 

You might find it easier to pull down the cove molding, overlay drywall, tape and paint, and reinstall the molding (or just tape the corners as normal).