I think you should certainly apply, as long as you have no personal opposition to Catholic education writ large; the previous question you link is a great source if you are uncertain.
As to your question, for an example, I pulled the mission statement for Marquette University in Milwaukee, WI.
Marquette University is a Catholic, Jesuit university dedicated to serving God by serving our students and contributing to the advancement of knowledge. Our mission, therefore, is the search for truth, the discovery and sharing of knowledge, the fostering of personal and professional excellence, the promotion of a life of faith, and the development of leadership expressed in service to others. All this we pursue for the greater glory of God and the common benefit of the human community.
Following this broad statement, they have four subsections on Excellence, Faith, Leadership, and Service. Of course the mission of your university may differ but I presume it will paint similar broad strokes.
I would write a statement that addresses the primarily non-Faith elements of the mission. Quoting again from Marquette, on Excellence they state:
education must encompass the whole person: spiritual and moral as well as intellectual, the heart as well as the mind
Without aligning with their religious beliefs, you could certainly contribute to education encompassing the whole person, particularly in intellectual pursuits.
Most U.S. Catholic universities that I am familiar with belong to either the Jesuit or Franciscan orders, though there are others as well - it might be worth spending a bit of time researching the order of the university you are applying to. Painting in a very broad brush, Jesuits tend to value intellectual pursuits as valuable contributions to society, and lean quite a bit more liberal than official Church policy on social issues. Franciscans are particularly concerned with service to the poor, and as such their universities try hard to accommodate students with financial difficulties. Even if you disagree on the religious aspects, you can probably find something in the general mission of these orders you also support.