となり can mean “next to” or it can mean a neighbor. While no is of course stating a possession on となり. You can see this type of pattern in other shows and movie titles (e.g. となりの関くん which can be translated into “My Neighbor Seki”).
This is a food that usually has something in the middle wrapped in rice and seaweed wrapping at the bottom (like so). Basically it is a sandwich but it uses rice instead of bread to hold the contents in.
If you grew up watching Pokemon you may have gone up to your parents asking for jelly donuts.
Unfortunately, I don’t think there is a good enough translation for this word so it is usually kept as Onigiri.
This kanji may look familiar to you if you have seen the kanji 君(きみ). They are the same, however it takes on a different meaning here. When put together with a name it gives off a honorific tone such as ちゃん、さん、様（さま）. It is usually thought of as the boy equivalent to ちゃん. This also doesn’t have a direct translation so it will be kept as is.
So put together we have “My Neighbor Onigiri-kun”. This title comes from a few shorts made by Studio Ghibli, who also made “My Neighbor Totoro” （となりのトトロ）.
Looks like they were taken off of YouTube. However, if you are interested in watching these shorts you can find them on the web.