Citizen Sleeper

There doesn't appear to be a way to win a ticket via the Draw after preparing the Sidreal. Later once the Sidreal is docked in the Hub, you can scout it and run into an old friend who will have a fairly quick mission to retrieve some data from a small ship in the shipyard. After doing some repeatable actions you will have one more critical action to retrieve the data. After this you will have an active scene with Lem and Mina in which the old friend will give you the tickets for Sidreal.

Note: There is a diverging path in the game in the dialogue choices directly after receiving the tickets. See 3 Body Problem guide for more info.

Short version: Complete Castor's questline to get a Sidereal ID (or multiple IDs).

Long version:

This is unlocked as part of any of the Sidereal ending/achievements (Solo Ticket/Left Behind/Long Journey/Three Body), so I am just copying and pasting from my solutions to those.

Dialogue choices do not matter for this achievement. (A few do affect which Sidereal ending you will get though.)

Go to the Shipyard and complete the Yard Hand wheel through either "Assist a Shipbuilder" or "Haul Materials". You'll then get a story bit where you're introduced to Lem & Mina. Choices here don't matter.

Go to Unit Assembly and complete the Junior Tech wheel by successfully completing "Systems Fitting" enough times, or complete the Welder's Mate wheel by successfully completing "Welder's Mate" enough times. (If you get a negative outcome 3 times, you'll fill the Quality control wheel and have to do the Shipyard portion again.) You'll get a story bit with Lem & Mina where you have to watch Mina while Lem works; again, dialogue choices don't matter.

Lem & Mina's Unit will unlock in Lowend, with babysitting Mina being an option. Babysitting Mina or not does not matter for this achievement, but does affect which Sidereal achievements/endings are available.

Go to Sidereal Bridge and complete Interface action "Sidereal Fit-Up" enough times to fill the "Sidereal Horizon" wheel (24 sections total; positive outcomes count for 2 sections). After this is done, go through another dialogue with Lem (choices don't matter). A 3-cycle countdown "Sidereal Draw" will start on Shipyard, and a 6-cycle countdown "Moving the Horizon" will start at Sidereal Horizon. Keep yourself alive and healthy until the 3-cycle countdown finishes.

Go to the Lem & Mina icon at the bottom of the Eye to go through a story bit where - surprise - you weren't even eligible to win the tickets. Continue to wait out the 6-cycle “Moving the Horizon” countdown. Depending on your choices, you may need to go through a dialogue with Lem after this countdown. Regardless, take the Ascender car up to the central hub, where there's a new icon for Sidereal Dock.

Warning: the next step will start an 8-cycle countdown timer; by the end of this timer, you will need to complete the next two sections below. Not hard, but just make sure you're not also chasing a bunch of other quests at the same time.

Repeat Intuit action "Scout the Dock" enough times to fill the 6-section "A Way On Board" wheel. After completing this step, you'll have a dialogue with Castor, who makes you an offer to get you onto the Sidereal. This starts an 8-cycle countdown; you will need to complete the section below before this timer runs out.

Go to Celis Ship at the bottom of the Eye. Repeat Engage action "Bluff Past Security" and/or Intuit action "Observe and Explore" enough times to fill the "Celis Infiltration“ wheel. Avoid negative outcomes as they will fill the negative "Celis Lockdown" wheel. After this is complete, go to Celis Servers and do critical action "Extract Celis Data". (A negative outcome still results in getting the data, but you'll lose 2 points of condition.)

Return to Castor, where you'll give him the data and he'll give you a Sidereal ID. Depending on your choices, this may be followed immediately by a dialogue with Lem & Mina, but your choices do not matter for this achievement. After this dialogue finishes, the achievement should unlock.