Im doing the same job right now.
http://www.miata.net/garage/top/You should be able to replace the cables with the top in place.
If you open the top half-way see if you can peel back the fabric from the passenger or drivers side near the front to reveal the rivet that holds the plastic connector that is on the front end of the tension cable.
This rivet would need to be drilled out in order to replace the cable. also try to peel back the fabric at the part of the frame that would be in contact with the rear of the window (window, not windshield).
Here you should find the cable connected to a tension spring and the other end of the spring screwed to the frame.
Make note of the path that the cable takes around the top hinges as you have to put the new cable in the same way.
If you have access to these two points then you can replace the cable without having to remove the top.
The only other step in the process is that you will have to fish the cable through the top itself. This should not be too difficult to do. You could tape the new cable to the old cable and pull the new cable through as you remove the old one.
Mazda part number for the soft top tension cable and spring is NA01-R1-215. (US Source)