I have fabricated, manipulated, and rigged a few shooting games in the last few years. Pics and actual name of the game would help.

This should be the assembly
Alangun stylus.jpg

Here is my '69 Spooks, probably identical to your '59 game.

I don't need to replace mine so I'm not taking anything apart, but looking at it:
To get to the stylus easily the first thing I would try is to remove the cross bar that holds the stylus bar from dropping down. It should swing down, exposing the stylus.

The stylus pin is mushroomed (swaged) to hold it in the bar as you can see, so I don't think the pin just pulls out.

At this point I would need to look at the new part and see what looks different. That would give me a clue to what to do in reverse to disassemble the stylus. For instance is the new pin mushroomed already, is it part of an assembly or just a loose pin?

The stylus assembly is riveted to the stylus bar so I would hope it doesn't need to be drilled out. Swinging the bar down may give a better look at how all this goes together. I'm sure they didn't design it to be easily replaced.
I think dropping the bar down will give a better idea how the new parts go in. Maybe they are pressed into the bakelite holder, or the rivets hold them on the bakelite holder.

stylus1.jpgstylus2.jpg

>>if you push from the mushroomed end will the stylus move freely as if it would come out except for the mushroom stopping it?

Yes, the mushroomed end goes all the way freely to the plastic base.
Does the new stylus have the contact pad already attached, it won't come off (the other end of the stylus pin)?

A lot of gunk builds up on the pin and may stop it from going all the way to the base.

Alan

It does look like the old pins just slide out if the mushroom is removed.
Functionally, you don't need to mushroom the new pins once installed. Pushing them down while sliding the arm over the board should hold them in until the next time the arm is removed (probably never).

Alan