There are many sources can help you to find the vacancies in Kuwait universities, but the essential way is to nock the universities' doors directly. Either you are looking for teaching job or administrative job, prepare your resume and dedicate a time to start your hunt.
Follow opportunties of The American University of Kuwait (AUK), it is one of the new private universities in Kuwait. It is offering degrees in Business, Art, and English. Visit the university website (link in the below resources section) and click on 'employment' of the top bar. The vacancies are classified to faculty positions, adjunct positions and staff position.
Check Box Hill College Kuwait it is private college for women, that provides studies in business, art and design. Go to the their website and scroll down to the bottom of the main page and click on 'career' on the bottom bar. The college is listing the current openings in the career page, select each job to see the requirements and how to apply.
See vacancies of Gulf University for Science and Technology (GUST), it is another private university has college of business administration and college of arts and sciences. To see the announced jobs on the university website, select 'administrative services' on the left menu and click on 'employment' under 'human resources'.
Consider openings in Australian College of Kuwait and Arab Open University - Kuwait, they are another universities on Kuwait that post their vacancies on their web sites, you can find 'employment' page or 'job vacancies' on the main page menus.