It's not that simple. To start with, once the cathode has been exposed to air, it's shot and the electron gun would need to be replaced. You would also have to replace the getter, bake out the tube under vacuum to remove moisture from the glass, then fire the getter to absorb contaminates. Next, frequently the air blast and debris from a sudden break will blast some of the phosphor off the face of the tube. Oxy-acetylene is not good for glasswork, normally an air-natural gas or air-propane flame is used. Glass is a very unforgiving medium to work with, if you don't heat it just right it will crack, and if it doesn't crack when you heat it, often it will crack as it cools or cool with high stresses giving it a tendency to crack or even shatter violently later.
Rebuilding a CRT is a complex, delicate and dangerous art under the best of circumstances, requiring a lot of specialized equipment. A high vacuum setup incorporating a diffusion or turbomolecular pump is an absolute necessity on top of specialized fires, induction heater to bake out the gun assembly and fire the getter, jigs and replacement parts for the tube itself. Rebuilding a tube that had its neck broken off is for all practical purposes impossible.