This one took a couple tries but isn't too bad if you plan ahead. I've listed the mission order below, but you should fast-track it if you can. Optional trade missions in {curly brackets}, recommended steps in [square brackets]. I started the Earth History Center by Sol4 hour 15, completed construction by Sol4 hour 31, and got the achievement at Sol5 hour 8.
*** Spoiler - click to reveal ***
[upgrade starting construction drone hub to level 3]
water pump
farm [melons]
wind turbine
mine [laser mine]
geothermal
outpost
nanite processor
mini storage
immigration center [recommend 2 of these as early as you can]
- Note: max your workers here for a while as you need 100 citizens in Sol 2 for an election in Sol 3, and 200+ citizens to for the final objective 'Survive one more Sol' to appear.
habitat
intake fan
trade hub
upgrade geothermal
grow melons
export melons
upgrade drones
build second drone construction hub
pick up a cargo pod from the map
storage depot
{export 160 rice}
battery
lightning tower
build 3 more buildings [makes sense to choose ones appearing next on this list]
[two more farms: one for Entari, and one for Xenosage]
upgrade a farm
research center
research Cerulean Glow (needs Entari and Xenosage]
{export broccoli} [later optional export missions are barley, and then barley again]
chemical plant
manufacture 200 Cerulean Glow
build another geothermal plant
bar & grill
build 4 more buildings [makes sense to choose ones appearing next on this list]
scrubber drones: this cannot be one of the '4 more' as it is dialog gated
police drones
retail outlet
hospital
mine 100 zorium
zorium generator
skyscraper
Earth history center
survive one more sol (once construction is complete)
You will want to do a lot more workforce management here, filling drone construction buildings, research lab, chem plant, etc to make them 100% efficient and get that mission task over with. I also placed my earth history center within range of three different level 3 drone construction units so I had 9 drones working on it at 100% efficiency.
Note: commenters have noticed that you need 200+ citizens in order for the 'Survive 1 more Sol' objective to appear and qualify for the achievement criteria, so I also added this to the solution spoiler notes above.
Commenters also recommended the following additions to the guide:
- build extra iron/copper mines and nanite processors. @StevesTurn had 3 at max level.
- you have to win a referendum by the end of Sol 3, so you need at least 100 immigrants in Sol 2