Hi Graham and a warm welcome to the forum.
You really only have two options for your masts, either using a lathe, purpose built or makeshift or doing it by hand. If you can use a lathe that is the easier way of doing it or, as has been mentioned, you can put together a makeshift device with a pistol drill, possibly even battery powered. I would then make some gauge rings which are of the correct internal diameter for various identified points along the taper to ensure that you have exactly the right taper. If you don't get that right the mast fittings will not locate at the correct heights.
If you are doing it by hand you could either make up the mast from square dowel, changing the diameter as you go down and building it up in thicker sections or start with a parallel blank. If you make a laminated mast it will actually be stronger and you will have less material to remove but you need to ensure that you have the steps in the cross sectional area at the right points. Sanding will be easier and the steps should have been designed such that when you sand the steps away you should be just about at the right taper.
If you have to make from solid then I would use a plane first to roughly remove the excess wood before getting the sanding block out. This is only like sharpening a pencil and if done carefully should leave you with little sanding to do.
A couple of things worth remembering though, I would make something to ocate the mast in while you work on it rather than have it rolling around on the work bench, half a section of plastic pi[pe works well here, and I would make yourself some curved sanding blocks by glueing sand paper to the inside surface of sections of plastic pipe. This will greatly reduce the flats on the surface as you sand.