I just started replaying the game and can safely say that the first two levels are pretty easy. They seem very difficult at first, but that's just because you haven't honed your skills yet. On your first play through, be prepared to die A LOT on the second boss, though. It probably took me 10 tries to kill the fucker.
Here's a few tips that'll make your life easier:
1. Block, block, and block. Counterattacking is one of Ninja Gaiden's requisites. Until you know what you're doing, don't go charging at an enemy with your sword swinging wildly.
2. There are several rooms in the first level filled with a couple white ninjas who may or may not have infinite spawn. Whatever the case, don't feel obligated to fight them each time you enter the room. After you've beaten them the first time to advance, simply run past them the next time you need to go through.
3. When you're fighting the first boss, don't use your stars, and don't try to pummel him repeatedly. Get a few quick hits here and there, then get the fuck out. He'll try to grab you or pull off a mad combo, so take advantage of your speed and mobility to avoid his punishing blows. If you've become comfortable with the walls, use them to launch damage attacks; otherwise, keep going in and out.
4. The second boss is way tougher than the first. Before you encounter him, make sure to stock up on elixers, arrows, etc because you will need them. When you enter his arena, you'll notice there are bunch of annoying monks who'll shoot energy balls at you--ignore them and focus on the horserider unelss you need health (which they predominantely drop). Your arrows will come in handy here, so try to send a few volleys his way whenever he's on the opposite end of the court. At close range, doing the jump + y technique is quite effective, and can also be performed off the wall where the monks tend to congregate (meaning you'll do damage to the boss and the monks, providing you with health).
Ninja Gaiden's all about persistence and using your head until your skills naturally improve. Like many games with old school gameplay, it demands near perfection and will severely punish you if your skills aren't up to snuff. Keep at it!
Denis Dyack needs to completely rip off the staff combat from NG.