If that's using a 18", 15 watt F12 (1.5" diameter tube), then you should use a FS-2 starter
You will need to cut out the old one, then wire in the new one. You can either install a quick swap socket for the starter (a $1 or so) or you can directly solder to the base of the FS-2 starter pins, or you could disassemble the starter can and solder the capacitor and neon tube inside to match the way it is in that. Of course open it is not as safe to have it open like that, and you really shoud just solder to the FS-2 pins and cover them with tape, or put in a socket.
Also you should meter the transformer BEFORE you put in a new tube. Compare the resistance to the transformer on a known good one, and they should be very close in resistance. from my experience, old transformers can start to cross-short inside and then reduce resistance and then putting in a new tube will burn it out in just 1 second with a nice super bright glow on both ends.