The only way you'll blow your CRT is if you tried to plug that dongle monstrosity into the speaker output of a power amp haha, guitars have a high impedance signal. Direct boxes actually lower the impedance, so that definitely won't help make your output safer. (Still safe) So I say go for it directly from the guitar, the worst that can happen is nothing, (which is likely) which probably means you actually do need to lower the impedance with a direct box. (Which I still doubt would work but who knows) An amp with a line out or a digital pedal board would be the most likely options for actually getting sound through.
Do you know what the tolerances are on connectors like VGA, coax, and bnc? My monitor has VGA and BNC, so BNC might be easier to use (fewer intermediate steps, more control due to separate sockets for sync, r/g/b, etc). I'm curious if you might know how high the voltage can go before I run the risk of frying something.
Also, my guitar is an acoustic-electric with a preamp, which would probably make a difference.
My background is in audio engineering & live sound, so I'm not sure about the specifics of those ports. But there is zero risk of frying anything with the signal generated from a guitar, even an acoustic w/preamp. You would need orders of magnitude more power to do any damage, like if you plugged the 1/4" inch into a power amp designed for passive speakers (which can take the same 1/4" input as a guitar for some stupid reason.) I have to imagine that the only interaction happening within the CRT is directly with the built in speakers, so even if you did that crazy thing the speakers would blow but the video might still work (kinda want to see what would happen here lol, we need one of those YouTubers who like to break things.) For you though, the worst that would happen is the signal getting distorted and sounding horrible. My guess is either port would work granted you have the right adapter, and I actually think you're more like to hear something from an acoustic because of the slight boost from the preamp. All this to say there's nothing to worry about, but it's a lot of work for something silly, I'm sure you could make the world's worst amp with it if you try hard enough, seems like it should work to me.
Oh, I think you misunderstand. I'm not sending the audio into audio, I'm sending audio into video. So the signal from the guitar goes into the video connector (there aren't any built-in speakers). Why? It might look cool.
Oh! Actually I've seen something similar done before with a modular synth! There's this awesome Instagram profile called Ellingson.tv with a guy that uses modular synths to manipulate CRTs and combines that idea with a "Pepper's Ghost" hologram effect. Highly recommend checking it out