Montauk is awesome. It's not what you would call a wild-style trout stream, because they stock it every night, and you'll usually be fishing with a few hundred of your new best friends. Even so, there's 3 miles or so of good water that sees minimal pressure. For this reason, it's usually considered the best of Missouri's 4 public trout parks.
Fly patterns will vary, depending on which stretch of the river you'll be fishing. The patterns will also vary depending on the time of the year, water temperature, etc. Real quickly:
The more popular flies include mohair leech patterns, zug bugs, prince nymphs, pheasant tails, glo-bugs (big egg flies), wooly buggers, crackleback dries, elk hair caddis, and gray scud patterns. Occasionally, they'll rise to good-sized stimulators as well. Go here for more information:
Missouri Trout Hunter -- Montauk State Park
Missouri Trout Hunter -- Montauk State Park Fishing Report
While you're there, you should also check out the Current River. The headwaters of the Current are what you'll be fishing at Montauk State Park. The difference is that the Current only allows flies and lures, and you're only allowed to keep one fish per day, and it must be at least 18 inches long. The fly selection is fairly similar to the park, but the further away from the park you get, the more you'll want to focus on natural flies rather than gaudy attractors. You'll also want to add some adams midge flies (#22-24) to your box for the late summer/early fall trico hatches. Traditional trico spinner flies sometimes come in handy, but I've always had better luck with the little adams. More information:
Missouri Trout Hunter -- Current River
Missouri Trout Hunter -- Current River Blue Ribbon Trout Area Fishing Reports
Hope this helps. Have a good trip, and catch a whopper.
aka Missouri Trout Hunter