NO SPOILERS
This is one of the harder achievements to earn in this game. I would recommend doing this after getting significant practice with flying the ship.
Tips:
-Practice flying in low orbit around the sun without crashing into it. (description below)
-Don't forget to use the "roll" mechanic on your ship to better orient yourself.
-Have patience and expect to take multiple attempts at this.
-I used a game pad to achieve this.
-Try to do this quickly as you will have limited time after starting a run.
Guide to Hotshot Achievement:
-Spawn and climb up the elevator to the ship. Take off and start heading towards the sun.
-Take a look at the sun and when you spot the sun station, "Roll" your ship so that the station is orbiting from the bottom of your screen to the top.
-Approach the sun at a medium speed, aiming for the top horizon. The sun's gravity will pull you both forward and downward, so in order to increase and change your speed, you really just need to use you upward and downward thrusters. getting close the the sun increases your speed, so just let that happen and start thrusting away so that you "miss" the sun and carry your speed forward.
-Keep doing this until the sun station makes a full orbit while heading away from you. You can quickly pan your vision down to see if it is coming.
-When the sun station starts to come underneath you, do your best to try and match it's velocity. This is a very tricky thing to do but just keep letting the sun pull you forward. If you use your forward boosters too much you may lose control and be unable to slow down and be ejected from the sun's orbit. If you miss it, just wait for another orbit and try again.
-When you get close the sun station's velocity (that is, the same speed and the same direction of travel) try and land on the bright orange teleportation symbol. Use your downward boosters to hold yourself against it.
-Carefully adjust your ship so that it is pointing across the station at the nearby open door. You must enter this door while outside of the ship in order to get the achievement. You can actually lock onto the door while in the ship and this will make it easier to enter.
-Take off and try to start approaching the door. When you can, lock onto the door and use the "match speed" utility to stay with it. Constantly try to move and match speed as you get closer and closer. It can sometimes help to also use the "Roll" mechanic to maintain your orientation. When you are in front of the door, try to orient so that the ship's bottom is facing the door and use your downward booster to hold yourself against it.
-When you are ready, try to exit the ship. do it quickly because sometimes you will not have gravity anymore while in your ship and the centrifugal forces will shove you to the ceiling. When you exit the hatch, be sure to still be locked on to the door. Match speed with the door outside the ship and gently float your way inside. Once you are in the achievement should pop.
Congratulations on earning the Hotshot Achievement!