If you arrive in May and there is a 3 month waiting period for OHIP, you will get it in August. You say your PR card expires in July so the answer is no, you will not have OHIP before your PR card expires. You can however apply for OHIP before your PR card expires, you just wont get it until August.
You should consider getting private insurance for emergencies and accident coverage in case something happens while you are waiting to get covered.
