As the description says, "without using a continue", so you can lose lives just not a continue.
To get through the game with relative ease, utilise the save feature, put the game on easy and give yourself unlimited throwing weapons (Choose 00 in the options and wait for it to merge into an infinity symbol).
Rescuing Naoko isn't a given and if you defeat the boss without rescuing her you will have to try again (make sure you have a save before the boss). When you are fighting the final boss, you have to prevent Naoko from being squashed by a falling ceiling. On the left and right of the screen there is a gap in the wall, throw shurikens in here during the fight to slow the progress of the ceiling. Now just defeat the boss without letting the ceiling descend the whole way.
I linked a video of someone beating the final level, credit to MegamanNG.