Another incredibly vacuous term that no longer has the same meaning it initially did, thanks to cultural theft and abuse by people who aren't really cowboys at all, but bored, middle-class white people. It used to mean to toughen up or approach a problem with a hard attitude, but now is used to refer to anything requiring effort, or anything in which people are competeing against each other.
People sport the logo on their clothing or their car because they think it gives them credibility as a cowboy or a redneck, since it is now cool to be associated with these groups once again. But when you get down to brass tacks, most fans of the cowboy up marketing scheme are those who have no connection with the cowboy way of life at all. See poser
, git er done
and white trash
Jenny thought that putting her "cowboy up" sticker on her SUV would make others think she was a country girl. In reality, most people just laughed at her.