srhaggerty:
Mountain Goats are very phlegmatic and can take a lot of lead. A lot of hunters consider that, for their size, they can take more lead than much bigger animals.
Remember that you may well get a shot at one next to, or very near to, a cliff of drop off so ideally you want to put him down as quickly as possible. A fall in the rocks could really mess up the horns.
The last goat I took was with a 300 mag and I got lucky. He was standing in a small grassy area about 200 yards from the safety of the rocks and cliffs. Because of concern for the cape I shot him behind the shoulder at about 150 yards. He just stood there. He didn't flinch. He didn't move. I shot him a second time and again he did nothing. It was as if I was shooting at a picture. Just as I fired the second shot I could see blood spreading and working down from the first shot. After the second, it was more blood, but no reaction. At the third shot he took a step or two and just fell flat. All shots hit within a couple of inches of each other and any one would have eventually put him down, but it took a combination of 3 before he went down.
Also, as mentioned, don't forget that the shot you get may be at a very steep uphill or downhill angle. If you don't have one, you might consider getting an ACI from "W". I use them on all my guns now and the ACI and some good software and drop tables and you're in business.
All that said and considered I would go with the 300 RUM and a well constructed bullet. I'm not saying the 25-284 wouldn't do the job, but if it was me, I'd want to hit him with a little more than the 25 cal pill.
Regardless of which gun you use, congratulations on drawing the tag and good luck with your hunt.