Sorry, you posted while I was looking for my turkey. Since then it was pictures, packing out, processing, then playing catch up at work. I missed the thread.
You said you understand the draw, so I won't spend much time on that. To get a muley tag as an out of stater you basically have to win a draw, AND a separate lottery. And no rifle hunting of muleys for non-res.
The bad news: Kansas has very little public ground. Where there is public land it is almost exclusively on/around BLM/USACE reservoirs. It can get pounded pretty good depending on the time of year and location. ...but there are some good deer to be found. Mule deer usually get pushed off the wildlife areas pretty fast. Also, permission on private land is reasonably hard to come by unless you know someone. It can be done, but you'd want to have it lined up WELL in advance. And in general people are much more protective of the muleys than they are the whitetail.
The good news: There is a lot of private land in the
Walk-In Hunting Area (WIHA) program in Western Kansas. Hundreds of thousands of acres. As you might expect, it's not all created equal when it comes to habitat, but you can find deer. I've chased some pretty good deer on WIHA out there in recent years, both whitetail and mule deer. There are several types of WIHA, some Sept.1-Jan. 31 and the majority, Nov. 1-Jan. 31. That can impact the timing of your hunt.
I use both fixed and climber tree stands when I can find a hedge row or creek drainage but if you're in muley country there is often a limited number of trees, so spot and stalk is the order of the day. There are places where a pop up would probably be handy, but I don't even carry one out there regularly anymore. If you're in the grasslands there's no way to brush it in, and if you're in cover thick enough to brush it in you can usually just tuck back into the thicket and be just fine.
The toughest part can be finding deer obviously. It's not like mountain mule deer hunting where you can gain elevation and then glass a lot of country from one vantage point. So lots of walking. Water hole hunting can be good early in the season. Our rut is usually Nov 6-26, or something like that. Rifle season starts the Wednesday after Thanksgiving and runs about 12 days, so you don't want to be there during rifle season. TONS of road hunting. The deer learn where they can and can't be seen from the road. You'll rarely see one of the big guys from the road during the season. Oh, and pheasant season starts the 2nd Saturday of November so that'll bring out a LOT of traffic. Either avoid it, or be in a deep timber funnel super early in the morning so that the bird hunters will shove deer to you, then hope you can get them to stop in time to shoot them because they'll be hauling ass.
In an ideal world I think you'd dedicate about 2-3 days to scouting and 7-8 days to hunting (longer if possible). Come up in very late October and drive the roads for 2-3 days glassing. If you happen to find a good deer on Sept.1 WIHA give him a run, if it's Nov. 1 WIHA then take good notes and be ready in a few days. Then Nov. 1 get after them. I usually spot and stalk hunt the mornings, then treestand hunt (or hunt funnels between bedding & ag even if I have to lay down in CRP grass) in the evenings, particularly in locations where I can see a long ways hoping to learn where a big deer is bedding or traveling back and forth to food. That'll help me hone in on him for the next morning.