Hi Herman and welcome to the forum.
Since you have a Gold Star, have a look around for the GS specific forum here as well. You might get more help if you post there although a lot of us watch more than one forum. Like me, and I have GS's too.
To answer your question, yes, the drive spindle should come out the bottom but, if the teeth have a sharp edged burr worn on them, (which is possible) the spindle could hang up in the case as you try to extract it. A diamond jewelers file works to take the burrs down.
I would have a close look at the old pump before splashing out for a new one. The originals are surprisingly durable and don't have to develop any real pressure, they just keep the lube flowing.
I notice you fly a 172. There are a number of pilots represented here. I used to own and fly a 1958 PA22/20-160 taildragger myself some years back. Great little plane and another great hobby, that is..... besides motorcycles.
Anyway, best of luck with your project.