There are two different Astronaut City Challenges to complete, 17 Aliens to Catch and 20 Districts to Conquer. If you are having trouble finding these challenges then be sure to use the Data Scan Red Bricks to locate them in the area.
The catching aliens challenge consists of activating an Astronaut puzzle point and then chasing down and arresting the alien with circle.
The Conquer a District challenges requires you to plant an Astronaut Flag usually on the highest point, or one of the highest points, in a district. To plant the flag you simply walk up to the spot and hit circle and once he plants the flag the challenge will be complete. For a list of all the districts in the game please refer to the Super Compuper Trophy/Achievement.
*Note* – On the map if you click on a districts information, triangle when highlighting a district, it will give you a breakdown of how many tokens, challenges, red bricks, and gold bricks are left in the district.
Here is a video walkthrough in case one is missed, or is hard to find:
Credit to packattack04082.