Rome Fiumicino–Leonardo da Vinci Airport: Take the Leonardo Express to Roma Termini: 32 minutes, about every 15 minutes, €14. From Termini, Monti is a 10–20 minute walk or one Metro B stop to Cavour. For 3–4 people, heavy luggage, or mobility needs, use the official white-taxi rank: the fixed fare inside the Aurelian Walls is €55.
Rome Ciampino Airport: Use Ciampino Airlink: a 10-minute terminal bus to Ciampino station, then a regional train to Termini. The integrated ticket is €2.70; allow 35–50 minutes including the connection. A licensed white taxi costs a fixed €40 to the Aurelian-Walls center.
Roma Termini: This is the station you want. Monti begins just south-west of it: walk 10–20 minutes depending on your address, or take Metro B one stop to Cavour.
Roma Tiburtina: Take Metro B toward Laurentina to Cavour or Colosseo, then walk. Budget 20–30 minutes from platform to hotel. Do not overcomplicate this with buses.