What a good-looking job! Nicely squared and shallow enough that you can't lose one jar behind the others!
As to primer, yeah, you may not need paint immediately, but you might want to do it sooner than later, especially if you cook a lot. It's so easy to say 'good enough for now' and let it go too long. Being right in the cooking area, you want to be careful that you don't start reaching for jars with cooking stuff on your hands. Get too much residue and paint won't stick, so it will all have to be sanded down and cleaned again before painting.