Appian Way

There's a street crossing just beyond the Quo Vadis Church and the visitor center (where you rent the bikes) that requires a great deal of care to keep from becoming a statistic, but once across vehicular traffic heading towards the catacombs and beyond is minimal. We walked it and enjoyed the pleasant stroll, but saw others on bikes as we proceeded on. The paving stones on the old Appian Way are the originals and present a rough surface for cyclists - most people ride off to the side on a well worn shoulder which makes for a slightly smoother ride.
The park website is here: http://www.parcoappiaantica.it/home/itinerari/appia-antica. Has some good maps and photos to get you oriented as well as providing a lot of good "before you go" information.

I have read that there is no traffic allowed there on Sundays and that is the best day to go

We haven't ridden but we have hiked a big chunk of the Appia - from Casal Rotundo all the way back into central Rome - and can say that parts of the road itself would be tough on a bike but there are narrow dirt paths on either side which would be far less bumpy.

Some photos here from the Parco Appia Antica website:

http://www.parcoappiaantica.it/home/itinerari/appia-antica

I can't vouch for the condition of the bikes, however. If you don't think you can manage the surfaces or would have an issue walking long distances should your bikes give out, then I would suggest not renting bikes from the park. That said, I'm not seeing a ton of super poor reviews from people who've rented them from the park recently.

You do have another option, though? Topbike Rental and Tours are seeing really good reviews, and appear to be endorsed by the Rick Steves guidebooks so you might want to look into them?

http://www.topbikerental.com/nuovosito/eng/tour-rome/ancient-appian-way-catacombs-and-roman-aqueducts-in-rome-italy

The section of road from Porta San Sebastiano to Cecilia Metella is the busiest, traffic-wise, and you can get around some of that by biking/walking the spur which passes San Callixtus catacombs.

Actually traffic is limited to taxis and buses on Sunday. That means you can actual survive walking between the city gate and the catacombs. Sunday is the best day to go.

Yep, Sunday is best but I wasn't sure if the OP has that option.

We had to brave it between gate and catacombs but managed fine with a couple of deep breaths and some wall-hugging.