We travel through FCO often and the terminals are open 24 hours. If you have a 6:10 flight then you're probably on the KLM to AMS. That flight will probably leave from Terminal B. Have flown that a few times!
You can walk over to the terminal about 4:15-4:25 AM as most KLM flights open their "desks" about 2 hours before a flight. We flew to Bulgaria over the T-day holiday and were at the terminal at 4:15 for a 6 AM flight and the desk opened as we arrived. (was not KLM)
Often on these EURO flights only one or two desks are open and lines can get long. Also NOTHING in the terminal area really opens until 7 AM so grab something at the hotel before you walk over!
Many friends of ours stay at the Hilton (on points mostly) to avoid the "commute" in the early AM!
Last week, we went through B Terminal and had to go through passport control after you clear security (that was NOT the way it used to be?)... Hopefully with a 5 month stay you have the documentation necessary. Often in the AM these guys are very bored, but sometimes they can be pretty observant.
Good luck!
Ciao,
Ron