The easiest stage
Out of all the stages in the game, this one will certainly give you the least trouble. The enemies, though numerous, aren't particularly strong, and most can be defeated with relatively little effort. Konami Man and Konami Lady should be able to get all the way to the end and rescue Goemon with little trouble.
You'll need to return
There are two important items in this stage that you won't be able to collect on your first visit. One is the Power Capsule from Gradius which increases the attack strength of everyone on your team. You can see it below the long platform leading up to the ninja castle where you are attacked by ghosts. The only way to get it is to have Konami Man or Konami Lady fly below the platform, and you can only fly if you collected the Konami Mantle from Hell (which you can only enter if you have King Kong). The second item is Goemon's special weapon, which is collected in the form of a lucky cat. You'll also need the ability to fly in order to collect this item. Go ahead and rescue Goemon, and leave these items behind until you collect the Mantle and can return.
Think of the dragons as bosses
In this stage, the toughest enemy that you will face are the dragons. Dragons are free to fly around wherever they like, and attack intruders incessantly. You'll need to master the technique to fighting these creatures if you wish to make it through the stage. One guards the key that you need to collect in order to rescue Goemon, and two more guard the rooftop-passage that you must cross on the way to Goemon's jail cell. When a dragon starts to fly in your direction, jump in order to lure it up higher than your normal position. Then start hitting it as quickly as you can, in hopes of striking it consistently enough to keep it from ever moving. If you manage to attack it quickly enough and freeze it in position, you can defeat it before it poses a serious threat to your health. As a reward, it will drop several items as opposed to just one like most enemies.