= Solution
Let $\mathfrak p\in\operatorname{Proj}T$. Some homogeneous $t\in T_+$ lies outside $\mathfrak p$. Choose $N$ with $N\deg t\geq d_0$. Since $\varphi_{N\deg t}$ is surjective, $t^N=\varphi(s)$ for some homogeneous $s\in S_+$; primality gives $t^N\notin\mathfrak p$. Thus $\mathfrak p\in U$, and $U=\operatorname{Proj}T$.
Fix a positive-degree homogeneous $s\in S$ and write $q=\varphi(s)$. The map
$$
(S_s)_0\longrightarrow(T_q)_0
$$
is surjective: after multiplying the numerator and denominator of any degree-zero fraction by a sufficiently large power of $q$, its numerator has degree at least $d_0$ and therefore lifts through $\varphi$. It is injective by the same device: if $a/s^n$ maps to zero, then $q^m\varphi(a)=0$ for some $m$, and after increasing $m$ the injectivity of $\varphi$ in degree $(m+n)\deg s$ gives $s^ma=0$. Hence the displayed map is a <ring isomorphism>.
The opens $D_+(q)$ obtained in this way cover $\operatorname{Proj}T$, and on every one of them $f$ is an isomorphism onto $D_+(s)$. The inverses agree on overlaps, so $f$ is an <isomorphism of schemes>. This proves the <Invariance of Proj under an eventual graded isomorphism>.
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