What type of cleat is this? Should I save it or get new cleats?
It's a Shimano SPD cleat. There are a few different models, but they are all cross compatible. You can reuse the old ones if you can unscrew it, but new pair should be under 20€, too.

EDIT: Looks like this is an old Shimano cleat, like this one: http://www.fixedgeargallery.com/reviews/dzr/ovis/10.jpg. Modern ones and most copies have a sharp angled "shoulder" on the middle section.