You can use Table and Rotate:
c = Cylinder[{{0, 0, 0}, {0, 0, .95}}, 1/8];
b = Ball[{0, 0, 1}, 1];
Show[Graphics3D[{Opacity[0.5], b,
Table[Rotate[c, angle, {1, 0, 0}, {0, 0, 1}], {angle, 0, 2 Pi,
Pi/3}]}], Axes -> True]
To remove the caps you can parameterize the cylinder:
c = With[{r = 1/8},
ParametricPlot3D[{r*Cos[t], r*Sin[t], z}, {t, 0, 2 Pi}, {z,
0, .95}, PlotPoints -> {20, 2}, Mesh -> None]][[1]];
b = Ball[{0, 0, 1}, 1];
Show[Graphics3D[{Opacity[0.5], b,
Table[Rotate[c, angle, {1, 0, 0}, {0, 0, 1}], {angle, 0, 2 Pi,
Pi/3}]}], Axes -> True]
