Contra

Simply work your way to the second tunnel without picking up any upgrades for your weapon. Relatively simple but the only advice I would give anyone going for this is don't bother killing any enemies that you don't have to. This mainly applies to the climb before the water tower and the second tunnel. Just keep on jumping. Ignore the mines and troops and turrets and wall mounted guns. Just run and jump all the way to the top and then use the same method you would for completing the tower for Bulletproof Jump up the left hand side and take out the first small turret, then land just to the right of the center of the tower to avoid the cannon balls. Then blow up the right hand side turret before focusing on the long thin turret in the middle. Once these are dont you shouldn't have a problem.

Also, stay clear of the fire circle that the first boss fires at you. It's not worth taking the risk of killing it and end up dead.

Hope this helps. If you need any clarification about the route to take then feel free to send me a message on here

Just rush through the game as quick as possible being careful not to shoot powerups because if you get any then you negate this achievement. I believe this can be done in multiplayer as well.