Most transcendental equation can't be solved analytically, we use NSolve
:
f[y_] := NSolve[-2 \[Pi] + 2 x + 2 \[Pi] y == Sin[2 x] && 0 <= x <= Pi, x]
f[1/2] (*For y=1/2 we have: *)
{{x -> 1.5708}}(* Pi/2*)
And Plot for range: 0<=y<=1
Plot[{y /. f[y], y}, {y, 0, 1}, PlotStyle -> {Red, {Dashed, Black}}](*Just executed this*)