"Can I stay airside and avoid immigration and just go through passport control."
Immigration and passport control are, in this case, the same thing. You cannot stay airside, because you are not on a "connecting" flight but two separate flights. Therefore you must "enter" Italy (and the Schengen zone) in the Rome airport before going anywhere else.
In April 2014 I did the same thing you are going to do, except I had checked luggage, and had three hours to make the next flight. I made it with time to spare (even though I had to claim my luggage at T3 and recheck it in T1), but of course the time it takes to get through immigration can never be predicted.
Since you will not have checked luggage, here's the procedure. You land at Terminal 3, then follow the herd into the terminal building (usually requires a shuttle train, as flights from the US land at a distant part of T3). You go through immigration, then make your way to Terminal 1. I did this outside and it took about 10 minutes; I don't know if there's a way to do this to avoid going outside (which would only be necessary if it's raining). Assuming you've printed your boarding pass, you then go through security and go to your next flight. You should be able to print your Alitalia boarding pass before leaving the US (I think you can do this 24 hours before your flight, so if you're flying at 10 AM on, say, June 15, that means anytime after 4 AM Eastern time on June 14).