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Yup, if the cap is no where near fitting I can almost guarantee you have an over extrusion issue. That can be from the filament diameter being wrong, the extrusion multiplier being wrong, or the extrusion width being wrong. What I would start with is checking out some videos on dialing in your extrusion multiplier. Makers Muse has some great info on this along with some "tolerance" test models so you can dial things in. It is worth getting it dialed in - if not for this, for your future prints! (https://www.youtube.com/watch?v=XnJBpCWf9Mc)
Also double check the measurements on that filament and see how close it is to spec. Is it the Prusa PLA, or another brand? Sometimes they will list the tolerance on there (like + or - 0.03) but if it is way outside that range, be suspicious.
As for printing the separate parts: from what I understand, in Cura you can just right click on the objects and click "Split object into parts".
If you can't get that to work, let me know and I'll post that part of the model as a separate piece!