There's no real reason to swap chassis boards without the neckboards unless you have a known bad neckboard that you don't want to repair. Properly dressing the focus lead for the voltage (~4-6kV) is not something I would recommend taking on for most people.
In general, if you've got a good tube from the same model monitor, it should be a straight swap as the yokes are compatible. Just pull the boardset (i.e. disconnect anode, neckboard, dag ground, and yoke) from one and hook everything back up on the other. Be aware that after doing this, you may need to adjust screen (aka G2), and color gains and cut-offs to even get a picture due to tube variation. Then do the standard alignment including focus, brightness/contrast, geometry, etc.
If you're not sure as to compatibility, check the yoke's inductance. The tube is very likely compatible as long as it has the same base (number of pins).
Standard precautions to pulling the anode apply: discharge it, wait a while, then discharge it again. Make sure you get everything hooked up right before you apply power. I like to create a checklist as I unhook things, especially on more complicated monitors that may have other things needing disconnection and hooked back up. Forgetting to hook up any one of the major connections is usually a major hassle (often resulting in damage) at best and a fairly bad safety hazard at worst (like leaving the anode unhooked, but that's pretty obvious).