Bus from Rome Fiumicino Airport
Quick answer: Several coach operators — Terravision, SIT Bus Shuttle, and Cotral — run airport buses to central Rome (mainly Termini or Tiburtina) in around 45–55 minutes. Published one-way fares are roughly €5–7 depending on operator and whether you book online, though it's worth checking each operator's current fare before travel.
Coach operators
Terravision and SIT Bus Shuttle both run frequent services to Termini, typically every 20–30 minutes through the day. Cotral connects to both Termini and Tiburtina and is generally the cheapest option. All depart from clearly signed bus stops outside the terminals.
Bus vs. train
The bus is usually cheaper than the Leonardo Express but slower and more exposed to traffic. It's a reasonable option if your destination is closer to a bus stop than to Termini, or if you're traveling with a tighter budget.