Zelda 1 - Walkthrough of Dungeon 1 of Quest 1: The Eagle (Zelda I NES mini)
Map of the labyrinth "The Eagle": Dungeon 1 of Quest 1 from Zelda I (The Legend Of Zelda NES mini)

Start by collecting the keys on the left (B6 and D6) and to the right of the starting point by killing the enemies (the 3 bats (Keeses) and one of the skeletons (Stalfos)). Then go up twice and kill all the Stalfos to get the key in C4 (1). Go right to pick up the compass. Go back left then go left again. If you want to leave from this room (B4), you will need to kill all the bats, but since you don't want to leave, go up (2). If you want to go to A3, kill the 3 "Gels" and push the block in any direction (3).

Otherwise, go right. Pick up the map and go up (1). Kill the Stalfos carrying the key (C2), take it and go up. Kill all the orange boomerang throwers (Red Goriyas), pick up the key (C1) and go left. Push the left block upward and take the staircase. In the underground passage, you will find the bow. Come back out. To pass safely, pretend to go through then backtrack: the spiked crosses collide and come back slowly (2). Pass through at that moment. Then go to D3 (to the right of where you found the map) and kill all the Red Goriyas to get the wooden boomerang (3). Go further right and avoid the hands that bring you back to the start to pick up the key (E3) and go up to face the boss.

The boss is called Aquamentus, a sort of dragon unicorn. To kill him, it's easy: just stick close to him and hit his head until he dies. If you have all your life points, position yourself as shown in image 2 and shoot until Aquamentus drops his heart. The door opens, go right to pick up the first piece of the Triforce.