May 1 is an excellent opportunity for a walkabout of the neighborhoods and sites which don't require tickets/hours. A couple of ideas:
Walk via Appia Antica: see the catacombs at St. Sebastian or St. Callixtus:
http://www.parcoappiaantica.it/en/testi.asp?l1=1&l2=6&l3=0&l4=0
Take the elevator to the top of Monumento Vittorio Emanuele
Wander the ruins at Portico d'Ottavia and past Teatro di Marcello - free and always open. There are signs in English to tell you what you're looking at:
http://www.turismoroma.it/cosa-fare/portico-d’ottavia?lang=en
http://www.turismoroma.it/cosa-fare/teatro-marcello?lang=en
Peer at Area Sacra at Largo Argentina from the sidewalks above (the ruins themselves are closed): Free and signs in English - or were when we were there. This is where Julius Caesar was assassinated:
http://www.060608.it/en/cultura-e-svago/beni-culturali/beni-archeologici/area-sacra-di-largo-argentina.html
Piazza del Popolo and excellent Santa Maria del Popolo:
http://www.060608.it/en/cultura-e-svago/luoghi-di-culto-di-interesse-storico-artistico/chiese-cattoliche/santa-maria-del-popolo.html
Basilica Santa Maria Maggiore:
http://www.060608.it/en/cultura-e-svago/luoghi-di-culto-di-interesse-storico-artistico/chiese-cattoliche/basilica-di-santa-maria-maggiore.html
Gesù
http://www.060608.it/en/cultura-e-svago/luoghi-di-culto-di-interesse-storico-artistico/chiese-cattoliche/chiesa-del-gesu.html
Santa Maria in Trastevere:
http://www.060608.it/en/cultura-e-svago/luoghi-di-culto-di-interesse-storico-artistico/chiese-cattoliche/basilica-di-santa-maria-in-trastevere.html
Basilica di San Giovanni in Laterano: the Pope's church as the Bishop of Rome (no, that's not St Peter's):
http://www.060608.it/en/cultura-e-svago/luoghi-di-culto-di-interesse-storico-artistico/chiese-cattoliche/basilica-di-san-giovanni-in-laterano.html
Basilica San Celemente:
http://www.basilicasanclemente.com/eng/index.php/informations/excavations
Walk up the Aventine to see the municipal rose gardens (in bloom!) and old. very interesting Santa Sabina. There's a great view of the back of the Palatine and other parts of the city from up there:
http://www.060608.it/en/cultura-e-svago/luoghi-di-culto-di-interesse-storico-artistico/chiese-cattoliche/santa-sabina-all-aventino.html
http://www.060608.it/en/cultura-e-svago/verde/giardini-ville-e-parchi-urbani/roseto-comunale.html
There are oodles of things to do - including more churches, parks, etc. A disclaimer here: I won't swear to the 100% accuracy of the hours provided on the web links - although that info is as 'official' as it gets. Churches can shut their doors for any number of reasons so don't hold me responsible if you find them so!
An additional note for those of you who will be in Rome on May 3: that is a free Sunday at all Italian state museums (in Rome: Borghese, Colosseum, Forum, Palatine, Castel Sant’Angelo, Baths of Diocletian, etc) and the most-visited of those will be a ZOO so it's another good time to run off in search of other things to do - although some of the lesser attractions may not be overrun. One RS member mentioned that she was unable to snag reservations for the Borghese that day (mandatory even on free Sundays) so they're already booked up. The complete list (Rome and surrounds) here:
http://www.beniculturali.it/mibac/opencms/MiBAC/sito-MiBAC/Archivio/Musei-in-evidenza/index.html?pagename=&id=557&special=&categoriaComunicato=&mycategory=&tipologia=&hid_mese=&luogo=Lazio%7CRM%7CRoma&cmdconferma=Ricerca
That may also be a good day to visit Rome's civic museums - which are not free on 1st Sundays: Capitoline, Markets of Trajan/Imperial Forum, and Ara Pacis.
http://en.museicapitolini.org
http://en.mercatiditraiano.it
http://en.arapacis.it/il_museo/editoriale