The arrest is already public record and was made public record the moment you were booked into jail. Many counties these days have web pages where one can search for someone through a jail search, bond search or criminal case search.
The only way to have this information redacted from the internet is to obtain a not guilty verdict and file a motion to have all of the aforementioned information expunged.
It will appear on your DOL records as soon as the police reports are received (and can be accessed in certain situations). It will appear on your Defendant's Case History as soon as charges are filed. Whether the arrest will be discovered immediately depends on who is looking and how diligently they search. Contact an attorney and explain your situation to get specific advice for YOUR situation, whatever that may be.
YOU must request the hearing within 20 days or you lose the right to a hearing and your license will be suspended.
The ticket itself is a public record, and as soon as the Clerk of the Circuit Court opens a file and puts it into the computer, anyone can find it if they know where to look.