Maven,
I always have used just regular twist bits. IIRC. I think the valve hole should be 5/16, maybe 1/64 smaller. Drill smaller than you need at first, and increase to fit by 1/64 increments. Maybe start 3/64 smaller than 5/16 then go up.
When you get to a snug fit but big enough for the valve to go all the way in, check out the fit of the nose of the valve to the antler. Does there appear any clearance there? If so, file or sand the high point down to where it is flush all the way around.
I have a sacrificial valve that I use to file/sand down to so the antler point will come down to blend to the valve. This is certainly an extra step that really isn't necessary, but it does produce a more pleasing appearance.
To drill out the center section of the tip, start small and increase the size as you go until you get to whatever size you want. Be careful you don't drill out the side. You can cap the hole anyway you want.
Now, it is time to install the valve. I typically use Devcon 5 minute epoxy, but you can use whatever you like. Taking a flat toothpick or just any old piece of wood about that size, mix up a dollop of epoxy (it is not going to take a lot) and spread the epoxy into the hole in the antler. Do not put it onto the valve body. Put it into the hole. I don't remember exactly why, but I unscrew the cap about a quarter of a turn and seat the valve where it bottoms out. I will hold the antler tip in an upright position for about 30 seconds while pressing down on the cap. Then set it aside with the valve end up for about 20 minutes while the glue dries. NOTE: take the tip and the spring out before starting the glue process.
I have found that if a person takes a little time here, it will result in a smother operating valve. I always polish the plunger with white rouge so that it is nice and shiny. It will make for a smother operating valve. It will also clean up any flashing on the hole drilled into the side of the plunger. Next, take a drill bit the exact size of the plunger and clean up the end hole in the valve body, this will clear out any glue that might have gotten into the hole.
As far as drill presses go, you don't really need one but it is a great convenience. I would also say that a bench vise really makes the procedure much easier.
Have fun putting this together. In my years of making these little things, I had made a drill index with the size drill bits that I needed in it and that really cut down the time looking for the bits. As a result, I have no clue what size bits I used. You will have to figure that out.