I'm going for the same class, and my OSO won't even let me start putting my package together yet. What he told me is that he has to get all the applications for occ 183 (summer) sent off to the board before I can start doing my PFT and ASTB. Basically you should be able to take the PFT and ASTB and get the other parts of your "book" together in April and May, then send off a package in July or so. Your OSO will probably want you to do the PFT more than once, including one right before your package gets sent in so you have the best possible score. You can improve your PFT scores pretty significantly with training (my maximum number of pull-ups has doubled since this summer, from 7 to 14), so while you're waiting the best thing to do is just PT like crazy, and study for the air test. Of course, your OSO might do things slightly differently than mine (they all have different styles), so you should talk to them and see what they want you to do.
"Peace on earth to men of goodwill. All others stand by."