It seems like you're interested in a unique and exciting way to explore Rome. "Bounce on and Jump off" might refer to a playful approach to sightseeing, possibly using a bouncy or fun mode of transportation to explore the city and jump off at various points of interest. While I'm not aware of any specific service or activity called "Bounce on and Jump off" in Rome as of my last update in September 2021, I can certainly suggest a few alternative options that could provide you with a similar experience:

1. Segway Tours: Segway tours are a popular way to explore cities while having a bit of fun. These tours usually take you to major attractions and allow you to "bounce on" a Segway as you navigate through the city. While not exactly "jumping off," you'll stop at various points of interest to explore and learn about the history and culture.

2. Bike Tours: Rome offers several bike tour options that allow you to pedal around the city, enjoying the sights and stopping at various landmarks. It's not quite bouncing and jumping, but it's an active and engaging way to explore.

3. Hop-On Hop-Off Bus Tours: These tours offer a similar concept to what you mentioned. While not exactly bouncing, you can "hop on" a double-decker bus, ride to your chosen attractions, and "jump off" when you want to explore. This provides flexibility and a great overview of the city.

4. Electric Scooter Rentals: Many cities, including Rome, have electric scooter rental services. You can "bounce on" an electric scooter and cruise around the city streets to your preferred destinations.

5. Walking Tours: While not as physically bouncy, guided walking tours allow you to "jump off" the beaten path and explore the city's nooks and crannies on foot. These tours often provide a more immersive experience.

Before you plan any activities, I recommend checking with local tourism websites, travel agencies, or recent travel forums to see if there have been any updates or new attractions since my last knowledge update in September 2021. Always make sure that any activities you plan are safe, reputable, and align with your preferences for exploring the city.