Hi Shellbell, welcome out west. I had to look up what the term Dude Ranch meant. And it turns out I've been to one in Utah called Zion Ponderosa, though I don't remember it being a working ranch with cattle. I did do a horseback ride though. Was a great place with friendly staff and lots to do.
Can't really help you anymore on that front. The one you have listed is in Antimony which is in between Bryce Canyon and Capitol Reef, about 3 1/2 hours south of SLC. Both areas could be explored in the time you have allotted, as well as Zion National Park. Again it all depends on how much driving you would like to do and if you want to be on the go.
If you choose to stay in Salt Lake, there is a bison round up on Antelope Island in the Great Salt Lake in October, which would be worth participating. It's about an hour from downtown SLC.
Salt Lake in mid-late October is the shoulder season. Most of the fall colors be gone and the mountains probably won't have much snow except for the higher elevations, so skiing isn't a likely option. It might be the ugliest time of the year, to see the mountains until they are blanketed in white. Having said that, you could spend a week hiking in Big and Little Cottonwood Canyon and others nearby, if you are into that. Park City is 30 minutes away and you could spend a couple of days up there too.
Here are a couple of event calendars you can adjust the filters and see if anything of interest is going on while you're here.
I might go down to Southern Utah's National Parks and the dude ranch for 4-5 days and then spend 2-3 in Salt Lake/ Park City.
Look at some photos of the places, maps and drive times, and see what is of interest and in your budget and a more concise itinerary can be sculpted.