First off I wanted to say that you most likely will be flying on your first day, so bring your headset! I made sure to have all of my zero-timers fly on the very first day because I think it is important to start out with something fun and memorable.
So a typical day at ATP will be made up mostly of self study. Each event that you will have with your instructor will have reading assignments and quizzes, which will take you about 3-4 hours to complete. On top of that, you are expected to expand on the topic a little bit on your own. That means researching on the internet, watching youtube videos, reading articles, asking fellow students/instructors, etc.
All of this will be in preparation for your syllabus event, which will be a flight/sim or ground lesson. These events are usually around 2 hours, with 15-20 min pre and post briefs for flights and sims.
Now as you progress through the program, your daily schedule will change a little bit, but this is the rough template of your schedule.
With regards to the writtens, definitely get them out of the way. I will add to Adam and say that they are not only dated, they have absolutely no educational benefit. The standard method of preparing for these exams is to memorize the answers. Literally, the prep books/apps are based on memorizing the answers to all the questions in the question bank for the specific exam, and that goes from the private written all the way up to the ATP written. So since you have some time until June, do yourself a favor and get them out of the way.
Let us know if you have any more questions.