It is so kind and humane that you are caring for this orphan cottontail who had no mother to protect it, feed it, and teach it the ways of the wild. It would certainly have become hawk or coyote bait within hours without your intervention. In this special circumstance, in which the cottontail was all alone, you saved him! Bravo!
It's been several years since you asked the question, so I hope the cottontail is still with you and has free run of your home. They have such gentle spirits. I had a similar situation and had a cottontail sharing my home.
He would come and go as he pleased through a bunny door, but most of the time he chose to stay inside. Such a treat!
Cottontails are naturally quite skittish, but given complete freedom to retreat to one of his favorite hiding places whenever he felt the need, he happily co-existed with me and my domesticated rabbit.
He was sweet, gentle, and fascinating to observe. He didn't eat nearly as much as the domesticated rabbit -- he was so small. And he actually learned to use a litter box -- he wasn't perfect about it, but was very good about only urinating in his box and on one towel I had set up for that purpose.
Since you probably fed your cottontail by syringe he might have become accustomed to you holding him and petting him as he matured. If he didn't, that's quite natural considering his heightened flight instinct. He still might eat small pieces of banana from your hand, investigate you if you sit quietly near him, and feel comfortable enough with you that he'll eat in your presence, groom himself, and even nap.
The cottontail in my home was quite skittish so would run into hiding if I made sudden moves or twitched. And he usually scampered into hiding after any short period of socialization. It seemed he only liked to socialize for a short while, then needed to retreat and snooze.
He groomed himself very well -- he was always very clean.
Cottontails can establish relationships with domesticated rabbits, touch noses as a greeting, rest in their proximity, play and chase them, groom and be groomed, and learn domesticated rabbit behaviors. Not all will do all of these things, but they can socialize with, and trust domesticated rabbits.
The cottontail who had taken up residence in my home ate the same food as a domesticated rabbit: kale, green leaf lettuce, arugula, living wheat grass (nibbling blades from the growing plant), romaine, spinach, carrot tops, sweet bell pepper slivers (no seeds), cherry tomatoes (cut in sixths), dandelion greens and flowers, carrot slivers, baby bok choy, fresh long blades of fine fescue grass, timothy grass rabbit kibble/pellets, Kaytee sesame krunch-arounds, dried papaya bits, fresh banana, pear (no seeds), blueberries cut in quarters, cherries cut in sixths (without the seed), plum. He didn't care for cilantro, italian parsley, raspberries, or apples. He drank water from a small ceramic bowl.
He used a litter box that had a bowl of kibble and a few krunch-around or papaya bits as treats to encourage him to visit the box. He also used a towel that had a bowl of water, and a plate of kibble so he could eat and go at the same time.
Outdoors, he was more nervous and, after a few minutes, would usually bolt back indoors to safety.
Similar to a domesticated rabbit, he snoozed all day and was more active at dawn and dusk.
I hope this helps a little, and I hope your cottontail is happy and healthy!
p.s. As with Saber's Eastern Cottontail above, our cottontail loved to snooze under our beds. He had spots under three of them, each for a different time of the day/night.