If $R$ is a graded ring then $R$ is Noetherian if and only if $R_0$ is Noetherian and $R$ is finitely generated as an $R_0$-algebra.
If $R$ is Noetherian then $R_0 = R/R_+$ is Noetherian and $R_+ $ is finitely generated, but for the converse I don't know what to do.
Maybe I need to use the following: A subset of homogeneous elements generates $R$ as an algebra if and only if they generate $R_+$ as an ideal of $R$, but I don't know how.