The DIRECT treni-labelled Leonardo Express- have only 1st. They run ev 15/20 mins.
Besides these there are normal local double-deck trains, most very new, doing all stops and NOT serving Termini-but many other stazione in Roma, these have 1st and 2nd and charge normal fares.
IF you hold a pass with passdays and will stay in eternal city Roma-just buy when there, will cost less as passday.
As I incidentally just passed there last week: it is a VERY long walk from air-terminal to treni-via a kind of skybridge and as airport itself seems under complete rebuild, its also mostly bus from plane to exit. It took me nearly 1 hr after plane stopped till reaching the treni-and that all without wait for luggage or special checks (domestic flight in). So do not plan too tight.