Hi,
first, I would ask the previous owner about his diet. Don't change food too fast, they need some time to adjust and to grow the right bacteria to be able to digest new stuff. If he isn't used to it you start with a few leaves of greens and increase that slowly for, say, a week.
Could the sticky poo just be cecotrope?
Hay is good, wouldn't give too much pellets for now, half a cup over the day. Also be sure that he drinks - some buns who are used to bowls can have difficulties to find out how to use bottles or vice versa.
I would wait with grooming him, if he's not used to it it may stress him additionally. Give him some time to settle in.
He sure looks chubby, but new zealands have a stocky build too.