I'm trying to remember a cartoon from the 1980s involving a cowboy and aliens. (No, it has nothing to do with the film Cowboy & Aliens.) There was a horse who could stand on its hind legs like a human. I'm sorry for the lack of details.
This is BraveStarr, which aired from 1987 to 1988 and which was produced by Filmation Studios. The cowboy in question was Marshal BraveStarr. It took place on a distant planet called New Texas. There were plenty of alien species around.
What gave it away for me was the horse — it's good you mentioned that detail. BraveStarr's horse was named Thirty/Thirty, and he could stand and walk on his hind legs: