= Solution
In coordinates $(ct,r,\phi,z)$ the diagonal <metric tensor> is $\operatorname{diag}(-1,1,f^2,1)$, and only $g_{\phi\phi}$ depends on position. Substitution into the connection formula gives
$$
\boxed{\Gamma^r{}_{\phi\phi}=-ff',\qquad
\Gamma^\phi{}_{r\phi}=\Gamma^\phi{}_{\phi r}=\frac{f'}f,}
$$
with all other <Christoffel symbols> zero away from singular axes.
Use the explicit <Ricci tensor> formula and the <Einstein equation> sign printed on the source formula sheet. They give
$$
R_{rr}=\frac{f''}f,\qquad R_{\phi\phi}=ff'',\qquad
\mathcal R=g^{\alpha\beta}R_{\alpha\beta}=2\frac{f''}f.
$$
For example, in $R_{rr}$ the two terms proportional to $(f'/f)^2$ cancel. For $R_{\phi\phi}$ the derivative of $-ff'$ contributes $(f')^2+ff''$, the product terms contribute $-2(f')^2$, and the determinant term contributes $(f')^2$.
\b[Formula-sheet qualification.] Directly contracting the separately printed <Riemann curvature tensor> as instructed on that sheet instead gives $R_{rr}=-f''/f$, the opposite sign. For the concrete example $f(r)=\ell\sin(r/\ell)$, that contraction gives $+1/\ell^2$, whereas the explicit <Ricci tensor> formula gives $-1/\ell^2$. The two source formulas cannot both hold with the stated contraction. The calculation here uses the explicit <Ricci tensor> formula, whose pairing with the negative <Einstein equation> coupling yields the requested <stress-energy tensor>.
The desired matter tensor assumes $\Lambda=0$; otherwise the cosmological-constant contribution must also be included. With that assumption,
$$
T_{\alpha\beta}=-\frac{c^4}{8\pi G}
\left(R_{\alpha\beta}-\frac12g_{\alpha\beta}\mathcal R\right),
$$
and raising both indices gives
$$
\boxed{T^{\alpha\beta}=\frac{c^4}{8\pi G}\frac{f''}f
\operatorname{diag}(-1,0,0,1).}
$$
The static observer measures <energy density> $T^{00}=-c^4f''/(8\pi Gf)$ and equal-magnitude negative axial <pressure> when that energy is positive. The axial stress is a tension. Keeping the source's curvature sign and its negative Einstein-equation coupling together is essential to this result.
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