No. I mean, you've just given the definition of a kernel. You need to apply it to this case. It's just the set of matrices such that a-3b=0. So just stick that in set notation...

Okay, you have a ring homomorphism which maps into the integers. Basically, you are trying to show that the image is isomorphic to the integers (this is easiest, and equivalent). So, how would you approach this problem? You know that the image injects into the integers (because it is a subring). So, what is left to show?...

This is a exercise in looking at the image to find things about the pre-image/kernel. So, if is a prime ideal, then what does this mean about the integers? If is a maximal ideal what does this mean about the integers?