There are two questions here:
1) One is asking if $\cdots\oplus R_{-1}\oplus m_0\oplus R_1\oplus\cdots$ is an ideal in a $\mathbb Z$-graded ring $R$, where $m_0$ is a maximal ideal of $R_0$, and the answer is negative as shows the following example: $R=k[t,t^{-1}]$, and (necessarily) $m_0=(0)$.
2) The other is an exercise in Marley's notes asking to prove that any homogeneous and maximal ideal $M$ in a $\mathbb Z$-graded ring $R$ has the form $M=\cdots\oplus R_{-1}\oplus m_0\oplus R_1\oplus\cdots$, where $m_0$ is a maximal ideal of $R_0$. In this case $R/M$ is a graded ring and also a field. This shows that $(R/M)_n=0$ for all $n\ne 0$ (why?), that is, $R_n=M\cap R_n$ for all $n\ne 0$. If $m_0=M\cap R_0$ it follows that $M=\cdots\oplus R_{-1}\oplus m_0\oplus R_1\oplus\cdots$.