About ten years ago, I thought I wanted to work in academia. Either in Admissions/Advising/Registration, etc. From my limited research and about a two year span applying and interviewing for non-teaching positions at public and private universities, my takeaways are this: 1. Have a degree from the school you want to work for. 2. Many seem to start at as part-time RA's working in dorms with students. 3. It's more who you know than what you know. 4. The pay sucks. 5. They don't prefer white males. YMMV. Good luck.
My first job offer out of Auburn was as a recruiter/admissions counselor for Loyola University in New Orleans covering the Southwest. It was to much travel and the head of the department rubbed me the wrong way so I turned down the offer. I don't think it matters what your degree is in. Write a decent cover letter and start sending applications. Focus on smaller schools as large state schools do tend to hire their own graduates. There are also professional groups on LinkedIn you can join to network/make contacts in the area of higher ed. Good luck.
Here's a job ad I found by plugging "university admissions counselor" into google. Human Resources - Post University Cory - I like your sig line.
First you should have good communication skills and if you can speak 2-3 languages than it would be and added advantage for you.