An apparatus for opening and closing an engine hood, suitable for a construction machine, in which an engine room is mounted on an upper swing structure and an engine hood is mounted on a cowl frame that forms the engine room to be opened and closed through a gas spring, comprising: a frame having one side fixed to the upper swing structure and the other side fixed to the cowl frame; characterized by a guide rail fixed to one side surface of the frame, and having a guide groove formed in a length direction and a plurality of seat portions formed to be cut so as to communicate in a downward direction with respect to the guide groove and maintaining an opening angle that is set to open the engine hood in multiple stages during driving of the gas spring; and a stay rod having a free end slidably formed along the guide rail and a fixed end rotatably fixed to the engine hood, wherein the guide groove formed on the guide rail comprises: a first sliding surface continued from a start end of the guide groove in which the free end of the stay rod is positioned to the first seat portion when the engine hood is closed; a second sliding surface formed between openings of the first and second seat portions (18, 19) with a height that is relatively higher than a height of the first sliding surface; and a third sliding surface formed between an end of the guide groove in which the free end of the stay rod is positioned and the opening of the second seat portion with a height that is relatively higher than the height of the second sliding surface when the engine hood is opened at a set maximum angle. a frame having one side fixed to the upper swing structure and the other side fixed to the cowl frame; characterized by a guide rail fixed to one side surface of the frame, and having a guide groove formed in a length direction and a plurality of seat portions formed to be cut so as to communicate in a downward direction with respect to the guide groove and maintaining an opening angle that is set to open the engine hood in multiple stages during driving of the gas spring; and a stay rod having a free end slidably formed along the guide rail and a fixed end rotatably fixed to the engine hood, wherein the guide groove formed on the guide rail comprises: a first sliding surface continued from a start end of the guide groove in which the free end of the stay rod is positioned to the first seat portion when the engine hood is closed; a second sliding surface formed between openings of the first and second seat portions (18, 19) with a height that is relatively higher than a height of the first sliding surface; and a third sliding surface formed between an end of the guide groove in which the free end of the stay rod is positioned and the opening of the second seat portion with a height that is relatively higher than the height of the second sliding surface when the engine hood is opened at a set maximum angle. a first sliding surface continued from a start end of the guide groove in which the free end of the stay rod is positioned to the first seat portion when the engine hood is closed; a second sliding surface formed between openings of the first and second seat portions (18, 19) with a height that is relatively higher than a height of the first sliding surface; and a third sliding surface formed between an end of the guide groove in which the free end of the stay rod is positioned and the opening of the second seat portion with a height that is relatively higher than the height of the second sliding surface when the engine hood is opened at a set maximum angle.