4×(1/2)^(1/3) = 3.174802" high.
Interestingly, it doesn't matter what the radius is. Let's call the radius r. Let's call the height the bartender should fill the glass h. It is obvious the radius of the half portion is the same ratio as that of the two heights. In other words, the radius of the half portion is r*(h/4).
Recall the volume of a cone is (1/3)*h*pi*r2.
Let's equate the half portion to half of a full portion:
(1/2)*(1/3)*4*pi*r2 = (1/3)*h*pi*(rh/4)2
(1/2)*4*r2 = h*(rh/4)2
2*r2 = h*r2*h2/16
2 = h3/16
32 = h3
h = 321/3 =~ 3.174802"