Cartilage piercings usually take at least a year to heal, and a lot of the time they take more. My first conch piercing took about two years and my second one took about a year and a half. My double helix was done over 3 1/2 years ago and it still acts up sometimes.
I'm a member of a forum where people write diaries of punches pretty regularly and it seems like people either say it didn't hurt at all or it was very painful. Some people get them punched and then stretched up a size and that's usually pretty painful, but it's different for everybody. Actually punches don't usually take much, if any longer to heal than a regular piercing, so you're looking at at least a year, but probably more. You'll know when it's healed. I considered my conches healed when they stopped getting crusty, were never sore, and I could comfortably change the jewelry.
I have 1/2" lobes and 11 piercings: left helix x2, both conches, both nostrils, septum, navel, and VCH.
My lobes are small but I know a lot about stretching and how to do it safely, so message me if you have any questions!