The phrase comes from the game of cricket. The wicket, as us Caribbean people well know (even the non-cricket fans) , refers to the rectangular area in the centre of the cricket field between the stumps. A bowler bowls the ball from one end of the wicket, bouncing the ball upon it, before the batsman hits the ball at other end. The wicket is usually covered in a much shorter grass than the rest of the field, making it more susceptible to variations in weather which cause the ball to bounce differently.
If the wicket is wet when the ball is bowled it will not bounce as high as usual, appearing, from the point of view of the batsman, to stick to the ground making it very difficult to hit the ball well no matter the shot played.
Hence a "sticky wicket" refers to a situation where there is no option you can take which is necessarily a good one. Such wickets are far less common in cricket now since matches tend to have stopped being played on uncovered wickets, especially in the professional sport.