To take down the Spider, you will need to move the right stick in tight circles on the Spider's body. From there, just repeatedly press
until you see the health bar at the top turn into a white bar with a skull in the center. Do not take the time to attempt to get a full lock before firing. This does not give bonus damage and will only slow you down.
The Spider should go down before you enter the last tunnel. Be careful of the rockets it fires as they come in from the sides and are not automatically targeted. There are also two enemies that appear right after you exit the first tunnel.
Good luck!