If you both like running, then I can strongly recommend Sight Jogging…
My wife and I ran with them early (7am ish) on the first morning of our honeymoon – early in the morning I'd recommend, first day of honeymoon less so!
It'd be just you, your partner and a guide. As I remember it, our guide was originally from California, had been a (normal not running) tour guide in Rome for 7 years, had recently set up her own pilates studio and ran marathons for fun. She talked the whole time and could've handled the pace been more than double anything we (well, me anyway) were capable of. They do all manner of routes, and it's neat feeling the flow of the ancient city while being caught up in the bustle of the modern one.
Just looked at their website again, and I don't remember it being that expensive for us. I guess after paying for the wedding everything seemed cheap! Whatever, we'll be going again when next in Rome!
Regarding restaurants, we had a fantastic lunch with the local business folk at Al Callarello. We stumbled upon it quite by chance and had a lovely afternoon there. As can be seen in this more helpful google maps image, it's just by the Baths of Caracalla.