How to plot this solid nicely?

ParametricPlot3D[
 Table[{r Cos[t] Sin[s], r Sin[t] Sin[s], r Cos[s]}, {r, 1, 2, .02}] //
   Evaluate, {s, 0, Pi/2}, {t, 0, Pi/2}, ImageSize -> Large]

Another Way

If we use the implicit expression of sphere, we can also construct the solids and it's complement by a relatively complex way.

SetOptions[ContourPlot3D, Boxed -> False, Axes -> False, 
  Lighting -> Automatic, BoundaryStyle -> None, 
  Mesh -> {{0}, {0}, {0}}];
f1 = x^2 + y^2 + z^2 - 1^2;
f2 = x^2 + y^2 + z^2 - 2^2;
f3 = x;
f4 = y;
f5 = z;
pureFun[f_] := (Evaluate[
     f /. {x -> Slot@1, y -> Slot@2, z -> Slot@3}]) &;
s1 = ContourPlot3D[f1 == 0, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, 
   MeshFunctions -> pureFun /@ {f3, f4, f5}, 
   MeshShading -> {{{None, None}, {None, None}}, {{None, 
       StippleShading[0.9]}, {None, None}}}, MeshStyle -> None];
s2 = ContourPlot3D[f2 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   MeshFunctions -> pureFun /@ {f3, f4, f5}, 
   MeshShading -> {{{None, None}, {None, None}}, {{None, 
       HatchShading[]}, {None, None}}}, MeshStyle -> None];
s3 = ContourPlot3D[f3 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   MeshFunctions -> pureFun /@ {f1, f2, f4, f5}, 
   MeshShading -> {{{{None, None}, {None, None}}, {{None, 
        None}, {None, None}}}, {{{None, 
        HalftoneShading[0.6, Orange]}, {None, None}}, {{None, 
        None}, {None, None}}}}, MeshStyle -> None];
s4 = ContourPlot3D[f4 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   MeshFunctions -> pureFun /@ {f1, f2, f3, f5}, 
   MeshShading -> {{{{None, None}, {None, None}}, {{None, 
        None}, {None, None}}}, {{{None, None}, {None, None}}, {{None, 
        HalftoneShading[0.6, Orange]}, {None, None}}}}, 
   MeshStyle -> None];
s5 = ContourPlot3D[f5 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   MeshFunctions -> pureFun /@ {f1, f2, f3, f4}, 
   MeshShading -> {{{{None, None}, {None, None}}, {{None, 
        HalftoneShading[0.8, Orange]}, {None, None}}}, {{{None, 
        None}, {None, None}}, {{None, None}, {None, None}}}}, 
   MeshStyle -> None];
Show[s1, s2, s3, s4, s5, PlotRange -> All]

The Complement

SetOptions[ContourPlot3D, Boxed -> False, Axes -> False, 
  Lighting -> Automatic, BoundaryStyle -> None, 
  Mesh -> {{0}, {0}, {0}}];
f1 = x^2 + y^2 + z^2 - 1^2;
f2 = x^2 + y^2 + z^2 - 2^2;
f3 = x;
f4 = y;
f5 = z;
pureFun[f_] := (Evaluate[
     f /. {x -> Slot@1, y -> Slot@2, z -> Slot@3}]) &;
s1 = ContourPlot3D[f1 == 0, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, 
   MeshFunctions -> pureFun /@ {f3, f4, f5}, 
   MeshShading -> {{{None, None}, {None, None}}, {{None, 
       StippleShading[0.9]}, {None, None}}}, MeshStyle -> None];
s2 = ContourPlot3D[f2 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   MeshFunctions -> pureFun /@ {f3, f4, f5}, 
   MeshShading -> {{{HatchShading[], HatchShading[]}, {HatchShading[],
        HatchShading[]}}, {{HatchShading[], None}, {HatchShading[], 
       HatchShading[]}}}, MeshStyle -> None];
s3 = ContourPlot3D[f3 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   MeshFunctions -> pureFun /@ {f1, f2, f4, f5}, 
   MeshShading -> {{{{None, None}, {None, None}}, {{None, 
        None}, {None, None}}}, {{{None, 
        HalftoneShading[0.6, Orange]}, {None, None}}, {{None, 
        None}, {None, None}}}}, MeshStyle -> None];
s4 = ContourPlot3D[f4 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   MeshFunctions -> pureFun /@ {f1, f2, f3, f5}, 
   MeshShading -> {{{{None, None}, {None, None}}, {{None, 
        None}, {None, None}}}, {{{None, None}, {None, None}}, {{None, 
        HalftoneShading[0.6, Orange]}, {None, None}}}}, 
   MeshStyle -> None];
s5 = ContourPlot3D[f5 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   MeshFunctions -> pureFun /@ {f1, f2, f3, f4}, 
   MeshShading -> {{{{None, None}, {None, None}}, {{None, 
        HalftoneShading[0.8, Orange]}, {None, None}}}, {{{None, 
        None}, {None, None}}, {{None, None}, {None, None}}}}, 
   MeshStyle -> None];
Show[s1, s2, s3, s4, s5, PlotRange -> All]

For the solid you can also use

pp1 = ParametricPlot3D[
   Table[{r Cos[t] Sin[s], r Sin[t] Sin[s], r Cos[s]}, {r, {1, 2}}], {s, 0, 
    Pi/2}, {t, 0, Pi/2}, ImageSize -> Large];

pp2 = ParametricPlot3D[
   Table[{r Cos[t] Sin[s], r Sin[t] Sin[s], r Cos[s]}, {s, {0, Pi/2}}], {r, 1,
     2}, {t, 0, Pi/2}, ImageSize -> Large];

pp3 = ParametricPlot3D[
   Table[{r Cos[t] Sin[s], r Sin[t] Sin[s], r Cos[s]}, {t, {0, Pi/2}}], {r, 1,
     2}, {s, 0, Pi/2}, ImageSize -> Large];

Show[pp1, pp2, pp3]

You can use SphericalShell with RegionPlot3D:

RegionPlot3D[RegionIntersection[Cuboid[{0, 0, 0}, {2, 2, 2}], 
   SphericalShell[{0, 0, 0}, {1, 2}]], 
  Axes -> True, PlotPoints -> 50]

RegionPlot3D[Region@SphericalShell[{0, 0, 0}, {1, 2}], 
 PlotRange -> {{0, 2}, {0, 2}, {0, 2}}, Axes -> True, PlotPoints -> 50]

same picture

Alternatively, you can use ImplicitRegion:

ir = ImplicitRegion[{1 <= x^2 + y^2 + z^2 <= 4, x >= 0, y >= 0, z >= 0}, {x, y, z}]

RegionPlot3D[ir, PlotStyle -> Red, PlotPoints -> 80, Axes -> True]