You would need to seal the tube to a vacuum manifold (the aperture of this should be a significant percentage of the overall size of the CRT) and pump it down (which means a diffusion or turbo molecular pump, and a rotary vane pump to back it). Vacuum is its own whole (very fascinating, somewhat expensive) thing.
There are some cathode coatings which can withstand air, such as those used on ionization gauges that may be exposed to poor vacuums and won't instantly fail even if briefly turned on at atm. Probably those are not the ones used in CRTs.
For people hacking stuff on their own with a box of scraps, I strongly recommend John Strong's Procedures In Experimental Physics from 1938 -- but I recommend you do not make your own blown-glass mercury diffusion pumps. The Fusor Forums are also a useful place for amateur experiments in this realm, even though they are focused, unsurprisingly, on fusion and fusors. Nowadays you can make some use of purpose-built vacuum hardware from ebay -- purchased new, this is $$$.
If you just want to build your own electron beam, this might be easier to do yourself, inside a large steel, copper, or glass vacuum chamber.
Electron beam lithography is used as a no-tooling-required method of extremely fine integrated circuit patterning. I believe the resists used with this are PMMA-based.
That monster with the carts going through it is firing the E-beam through a window made of very thin material (and also tremendously powerful). This device is used for cross-linking plastics "radiation cross-linked PE" wire insulation is made by this method.