I see a couple of problems

- Liquids in wood: some woods (red oak, for example) will just let the liquid pass through the end grain. And repeated wet-dry cycles in any wood can cause problems with rot and warpage.

- Differential expansion/contraction. Most woods expand twice as much tangentially (around the growth rings) as radially (from the center out). If they expand/contract at the same rate, all would be in synchrony, but since c=pi*d you'll have a lot of stress going on. This means as it dries (and wets and dries and wets and dries) the bottom is likely to make a nice pie-shaped split. If you ever look at the end of a piece of firewood, you'll typically see these cracks.

Dang that's some info I didn't think of! I just saw a mug on the show Vikings and I thought it would be pretty cool to have one that I made

If wood is the important consideration, and not just using a different material and laminating the outside of the cup with a veneer, you could try bamboo. Here's a link to an example... Clean, Chic Bamboo Cups

One problem that I foresee with the instructions on that page is that many waxes and sealants are alcohol soluble - and you know, beer! lol

You may do a bit of research, but maybe there's another type of wood that may also work, and of course if the sealant is required, one that will hold up to alcohol.

I don't know if you might be able to find something that would seal a cup you made but you did ask how to make a cup. You would usually need a lathe. I just googled quickly and came up with all sorts of diy lathe stuff. Here is one.

I didn't watch this one all the way, and as he said, you might not want to do things like he did. However, it could begin to give you ideas on how to proceed. The link on google would show all sorts of other links on the side for different (perhaps better) approaches. The basic idea is that you need a motor (perhaps from a trashed washing machine) and something to anchor the motor, and stuff to attach your piece to the shaft, and then a sharp tool (but probably not too pointed on the ends of the sharp part; a real tool would be a turning scraper) to cut the wood with a support to prevent the tool from moving up and down. A well placed block of wood would probably do the trick.

Anyway, watch a bunch of this stuff so you can begin to understand the problems and solutions so you don't end up blinded or slashed to pieces but might actually make something for some work and not much cost.

People have made and used wooden mugs and bowls for millennia, so it certainly can be done. For occasional use almost any finish will work assuming you don't let liquid stand in the mugs, e.g. oil (that would have to be renewed periodically) or polyurethane varnish. For a more durable finish I'd use epoxy, at least on the inside-- a thin, brush-on formulation like you'd use on a wooden canoe that will soak into the wood to some extent. That will effectively seal your end grain. Pretty much all finished are non-toxic once cured, so you don't really need to worry much about "food grade" finishes.