Never Tell Me the Odds - Complete mission without colliding with any asteroids. - 1500 Plex.
Cutting Through the Chase - Complete mission in 75 seconds or less. - 1000 Plex.
Nero suggested filling in the last Weapons Bay, but you can actually adjust any aspect of the Erichthonius. To see all the choices are available for each node, remove what's already there. Once you've picked a piece of equipment with an ability, you'll be prompted to bind it to a button.
After you've added the Photocaster Cannon and filled in the rest of the Veterancy and Corona nodes, the rest of the default equipment should be all right, unless there's a particular design goal you had in mind.
There's a button labeled "Info" at the bottom left corner of the screen (next to the Launch button). This will tell you the upcoming mission's bonuses, which can be helpful for figuring out what equipment you need.
You don't need to fight the enemies if you don't want to. Just rush past them with your movement abilities.