Em--it is probably related to filling out forms. Many doctor's offices have "Time of Arrival" on the sign-in sheet. Although I've not heard this particular usage before, to me it's no worse than "enter him", which, esp. if interpreted literally, sounds quite painful! In the latter, I believe everyone understands that it is short for entering his name, and (in the logbook or register) is understood.
In this case, possibly part of the duty once a patient has arrived may encompass other things; for ex., noting payment and/or putting a new form in the front of the patient's file for Reason for Today's Visit, etc. etc. Probably all of these kinds of things are encompassed in the meaning of "arriving" a patient. Mercy--as busy as doctors' offices are, if the staff said, "I have entered his name in the register and completed all the requirements for his arrival process" for every patient, they'd probably lose a whole appointment's worth of time by the end of the week!
