From someone who has never been, the only thing I might be able to offer is to soak in the golf, but also take some time to soak in the region itself. Whether that means stopping for a half hour in Mullen or finding spot along the Dismal River or McConaughy Lake for 20 minutes, I'd say do it.
The drive times are so long that you will feel crazy doing this, but at the same time...I did it myself. I drove all the way to Alliance, NE just to see "Carhenge", which is exactly what it sounds like. So I stood there in the dust for five minutes, went "Huh, so that's Carhenge," and then left. Fortunately, I discovered the Museum of the Sandhills, also in Alliance, which made the trip more worthwhile.
On the 80 corridor west of N. Platte, Ole's Big Game Steakhouse is a memorable restaurant (the walls are covered in hunting trophies) with un-fancy but above-average food.
Gothenburg, besides being home to Wild Horse, is an attractive small town with a Pony Express museum.
There's an enormous Cabela's in Sidney (corporate HQ is there, I believe) if you're into outdoors stuff. However, it's not on the way to Ballyneal--unless, that is, you go via Alliance!