First find a parameterization for the curve of intersection.
Given the equation of a cylinder, a natural choice for a parameterization would be one utilizing cylindrical coordinates. Here,
[tex]16x^2+y^2=16\implies x^2+\left(\dfrac y4\right)^2=1[/tex]
which suggests we could use
[tex]\begin{cases}x(t)=\cos t\\y(t)=4\sin t\\z(t)\end{cases}[/tex]
with [tex]0\le t\le2\pi[/tex], and we get [tex]z(t)[/tex] from the equation of the plane,
[tex]x+y+z=12\implies z(t)=12-x(t)-y(t)=12-\cos t-4\sin t[/tex]
Now use the arc length formula:
[tex]\displaystyle\ell=\int_0^{2\pi}\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dz}{\mathrm dt}\right)^2}\,\mathrm dt[/tex]
[tex]\displaystyle\ell=\int_0^{2\pi}\sqrt{\sin^2t+16\cos^2t+(\sin t-4\cos t)^2}\,\mathrm dt[/tex]
[tex]\displaystyle\ell=\sqrt2\int_0^{2\pi}\sqrt{\sin^2t-4\cos t\sin t+16\cos^2t}\,\mathrm dt\approx\boxed{24.0878}[/tex]
