To be honest, if it were me travelling, I would go with the 10:10 a.m. option from Termini: RV 4595, FR 9524 & R18213, departing on the Leonardo Express at 9:23 a.m. (or earlier). FCO is surprisingly efficient, especially given that it is in Italy, and an hour should be enough to get through immigration, collect your luggage and exit the airport. I would consider that comfortable, but I do live in Rome and am familiar with the airport and railway connections, and I speak Italian. You have to consider how much unfamiliarity with the lay of the land will slow you down—that depends on the traveller—but, IMHO, the biggest risk with my suggestion would be a significant delay in the arrival of the plane.

Since you're travelling in a few days time—you mention Sunday, so I assume it is this coming Sunday—it is most likely too late get the Super Economy fare that rove312 mentioned, and I imagine that the price of the Economy fare will be close to the that of the Base fare, so I suggest going for the latter, so that you have the flexibility to change to a later train should things go pear-shaped. (Note that you may have to pay a fare-difference, but at least you won't lose the value of your ticket.) That is what I would personally do, and the risk should be minimal. (If it appears that there will be a significant delay departing Chicago, then you need to make the change before you take off.) Of course, you have to take your personal circumstances and risk appetite into account.