You need thin, or extra thin CA, not normal medium CA, or it won't wick into the board and help hold it together, but mainly just stick to the surface.
I'd certainly tape up the whole neck and even the sides of the board, as the thin CA loves to run everywhere.
I used GlueBoost CA, as that's non-fuming (well a lot less so than normal as I occasionally got a few whiffs). Even so, it's best done outside for ventilation.