My gusse will be it gose in the carbi or atleast far enough to the carbi to get the plunger and slide on then you should be able to screw the plunger back into the carbi.
No you should not have to take the carbi off, undo the cap on top of the carbi (the throtle cable connects to the cap) then swap the plunger and slide onto the 1/4 turn cable.
I don't know as I don't have a 1/4 turn cable but thats my gusse.