So apparently a majority of people don't have the time of birth on their certificate, or they never realized it was there. Mine is clearly stated as 7:01am (that's right, I ruined my mothers day). I would suggest calling up the hospital and asking them if they still have a copy of your records. After x amount of time, they usually don't and they will tell you who you need to contact to get a new one, as well as all of the things you'll have to have in order to prove that you are you, and what you'll need to do to jump through the hoops for the new one.
While you have them on the phone, maybe ask them if your birth time should be included on that, or if it ever/always was. If they say no, and cannot help you in any way in that regard, maybe ask your parents if they are available about what time you were born, or if they recorded it in a baby book or maybe saved your hospital ID bracelet.
I understand the importance of knowing what time you were born. That's the difference between being in the house of Virgo or Neptune. Best of luck to you!