Hi, I'm going backpacking through Europe in July and I'll be staying in Rome for 4 days. I would like some advice regarding my itinerary, if it's possible to visit all of these places, and also where I can find cheap places to eat in each area. I'll arrive on monday morning.

Monday

• Morning: Arrival and Check-in, Piazza della Repubblica
• Afternoon: Trevi Fountain, Pantheon, Piazza Navona
• Evening: Piazza di Spagna, Dinner near Via del Corso

Tuesday

• Morning: Colosseum, Roman Forum and Palatine Hill
• Afternoon: Altare della Patria / Piazza Venezia, Largo di Torre Argentina
• Evening: Trastevere

Wednesday

• Morning: Vatican Museums and Sistine Chapel, St. Peter's Basilica
• Afternoon: Castel Sant'Angelo, St. Angelo Bridge
• Evening: Borgo and Prati

Thursday

• Morning: Villa Borghese, Pincio Terrace
• Afternoon: Circus Maximus, Orange Garden, Knights of Malta Keyhole
• Evening: Monti Neighborhood

One of my fears is to focus only in the touristic places and not get a "taste" of the city itself, is there something I should do instead of some of those attractions or I'll have the time to do it with this itinerary?