The big cattle ranches that use horses are more commonly found in BC not Alberta but there are some here south of Calgary more towards the foothills. Being a ranch hand pays not very well (I know because we employ a few hired men and I also work in the Ag accounting industry here in the Calgary area). Many ranch hand positions do look for people with a Class One Drivers to be able to drive the big grain trucks legally.
In the feedlot industry there are more jobs available, but again they do not pay that well. Many of the feedlots supply the horses used for penchecking now and do not allow you to bring in outside horses. As well, you need to have some cattle experience to get one of those positions as you are responsible for the health of those animals. I will also warn you from experience that the worse the weather the longer you are on horseback. My husband and I owned a feedlot and I did the penchecking. It can be a rough job to say the least.
If you do want to go that route though, most of the feedlots are located in South Alberta near Pincher Creek and Lethbridge, some in Nanton area, also Strathmore and Three Hills. Pull up local papers online to look for employment ads.
There are lots of horse barns/arenas in the Calgary area. Ranging from Jumpers to Cutting and Working Cowhorse.
Try the Northernhorse.com website for employment opportunities. Some get posted on there.