The length of time that you should spend studying for the DAT will depend on your background knowledge of the material, specifically the sciences. If you were successful in your science classes in undergrad and exposed yourself to several upper level science courses, you may not need to put as much time and effort into studying for the DAT.
Since your DAT is scheduled for the end of June, you can study as much as your work schedule allows and take a full length practice test as your test date approaches in order to gauge your progress. If you are not scoring as well as you'd like, make sure to reschedule to a later test date. Never take the real DAT just to see how you will do. You want to only take the exam when you feel confident in your knowledge of the material and have practice test scores reflecting your understanding of the content.
As general advice, try to develop a study schedule that incorporates both content review and practice questions, including some full-length practice tests. Good luck studying!
