This should still work. Can you select the multipoint feature and look on the table of contents List By Selection and make sure that the multipoint is the only selected feature? You will get that message if you have multiple features selected when you try to go to Edit Vertices. You can also double-click the multipoint with the Edit tool to enter Edit Vertices mode.
Once you are editing the vertices, click the Sketch Properties button on the Editor toolbar to open the window where you can edit the x,y,z,m coordinates.