I'm fairly sure I can get the old one out - but the rivets are stainless so any thoughts on that would be good.
Drill the heads off. I usually use around a 1/4" drill bit. Having a sharp bit makes all the difference. Once the head is off, use a hammer and punch to drive the shank into the mast.
Also, I was surprised to see the new casting does not have any holes, so I assume I need to drill those. Do I use a 3/16 drill bit for a 3/16 rivet?
3/16" bit or possibly one size larger like a 7/32". It doesn't have to be a really tight fit because the rivet expands as it's pulled. When drilling the holes, use the existing holes in the mast as a guide to mark the base. Then use a center punch so the drill bit stays centered on your mark. It will be easiest to drill with the base mounted in a bench vice.
Since I have never worked with stainless rivets before, is my cheap riveter going to work or do I need something special for SS?
The cheap hardware store rivet guns are designed for aluminum rivets. They will work, but you have to pull really, really hard. If you don't have strong hands, you won't be able to do it (and I highly recommend wearing safety glasses if you use one of these guns). Harbor freight sells a very nice high power rivet tool for about $25 that will pull stainless steel rivets easily.
Last, there is the question of electrolysis - it sounds like I need to coat the rivet in Lanocote or something?
Wouldn't hurt, but not absolutely necessary, especially if you sail in fresh water or rinse the mast off after sailing in salt water.