If you're motivated and resourceful, you can probably prepare for the test in a month. And applying after June is fine, granted your entire application is well-rounded. To put this in perspective, I submitted my app in August and got admitted last cycle. You can also submit your application before your test scores come in, although I would imagine that schools won't review your file until that score is uploaded. However, it takes time for the AADSAS to process the other parts of your app, so you may as well submit it as early as you can.
Back to your inquiry on the DAT, I would recommend browsing through some breakdowns to see how others approached the test, what resources they used, and how they scored.