In the last five years in New Mexico I've taken 3 goats and my son has taken one. When he took he he used the same gun as I used. All I've ever used there is my Remington 25-06, 115 gr., ballistic tip. The key is the ballistic tip. I swear by it for pronghorn hunts. It's a flat shooting bullet that I have no problem with it out to 400 yards. I know on my rifle it's dead on at 200, 1" low at 300. 3" low at 400 and 1" high at 100. I have never had to take a shot over 300. I took my second at 300, but my first was less than 75. The last was about 200. My sons was right at 150. I think you should be able to get within 300 or less on any goat. I've never had a problem and I'm a big guy so I not what you would call "stealthy". Try to stalk towards them with the sun at your back. Although they have unbelievable eye site, it's only eclipsed by their curiosity. If they are not sure what they are looking at they will sometimes stick around to find out what you are before taking off. Also get a large burlap cloth. We have gotten within 100-150 yards of goats with the sun at our back and walking behind a large burlap cloth. Like I said, sometimes their curiosity gets the best of them.
Give the ballistic tip a try at the range. I love it's accuracy and range and I think once you shoot it you will to.
Good luck. They are a fun hunt indeed.