There are a bunch of reasons why an old CRTs colours are a bit dull. First thing to look at is signal input, anything less than s-video from a console can look "washed-out". RGB or component give better results.
Washed-out colours could also be just the brightness/contrast/color settings accessible via your TVs menus.
If your colours are looking too purple/greenish, you can try using a degaussing wand to realign the phosphors behind the screen. However this probably won't do much to improve brightness.
Next stage of things to try, it could be worth pulling the back off your TV and tweaking the screen voltage pot on the flyback slightly. This will improve brightness and contrast, but don't tweak the flyback so high that you can see faint diagonal retrace ("flyback") lines near the top of the screen. After tweaking the screen voltage pot, you'll probably want to tweak the screen focus pot.
Finally, sometimes old CRTs have been used a lot and the cathodes in the RGB guns get worn out. This is a good time to find someone with a CRT rejuvenator, it might give your tube a few more years of good service. This is really the "Hail Mary" option, do it when you've tried everything else.
Also, as Bobby has already noted, your cab is unlikely to have enough depth to accommodate the CRT neck. So maybe stick with your LCD plan for the arcade cab and keep the CRT for console gaming from the sofa. Or build a second cab! Heh, once you start you can't stop.