Placing a contour line at y when f(x)>=y

You can add a mesh line using the options MeshFunctions and Mesh:

ContourPlot[0.5 (x y - 4)^2, {x, 0, 3}, {y, 0, 3}, 
 Contours -> {0, 1, 2, 3, 4}, 
 PlotRange -> All, 
 ContourShading -> None, 
 MeshFunctions -> {# #2 &}, 
 Mesh -> {{4}}, 
 MeshStyle -> Black]

The square term and it being the minimum throws it off.

I think your best choice is to just solve your equation for when it is zero separately.

Solve[0.5 (x y - 4)^2 == 0, y]
(*  {{y -> 4/x}, {y -> 4/x}} *)

Then plot and combine.

lp = Plot[4/x, {x, 0, 3}, PlotRange -> {{0, 3}, {0, 3}}, PlotStyle -> Black]

cp = ContourPlot[0.5 (x y - 4)^2, {x, 0, 3}, {y, 0, 3}, Contours -> {1, 2, 3, 4, 5, 6, 7}]

Show[cp,lp]

You can also generate just the 0 contour with the CounterPlot[ ] function, taking advantage of your knowledge of the problem. You can merge this with the rest of the problem as per above.

ContourPlot[ (x y - 4) == 0, {x, 0, 3}, {y, 0, 3}]