Ripley's shuttlecraft "Narcissus" floated through from the outer rim where LV-426 was, all the way into the core worlds. She's found, and taken to Gateway Station, a space station orbiting around/near Earth. Where she's interrogated and eventually finds a job.
But in those 57 years, technology has improved. Faster than Light travel allows for what was a 1~2 year trip to be around 2 weeks. Especially military crafts, who have better tech than old commercial ships. It takes a 4 weeks from the ship that finds Ripley to her arriving at Earth (either that ship takes her back, or a set of shuttles or transfers). But it only takes them 3 weeks to reach the LV-426 (renamed Acheron) from Earth, at a much greater distance.
The script itself also explicitly states 3 Weeks:
EXT. DEEP SPACE - THREE WEEKS LATER 25
An empty starfield. Metal spires slice ACROSS FRAME.
A mountain of steel following. A massive military
transport ship, the SULACO. Ugly, battered...
The timeline on wikia, gathered from the DVD specials:
May 16th - Shuttlecraft Narcissus is discovered by a deep salvage team, and Ellen Ripley and Jonesy are rescued.
June 8th - Ripley arrives at the Gateway Station and is placed into hospital, where she later learns what has happened.
June 12th - Ripley is discharged from hospital. At a board of enquiry, Ripley’s stories about the Alien are widely ignored. She has her flight status revoked, and takes up residence aboard Gateway Station, working with powerloaders in the loading docks. Following the enquiry, Carter Burke provides Hadley’s Hope with the co-ordinates to the derelict spacecraft on Acheron.
June 19th - Russ and Anne Jorden are sent investigate a set of grid co-ordinates provided by Carter Burke, beyond the Ilium Range on LV-426, with their children, Timmy and Newt.
June 21st - The Jorden family discovers the derelict with their children. Russ is attacked by a facehugger and impregnated with an Alien embryo. He is taken back to Hadley’s Hope.
June 26th - Xenomorphs swarm over the colonists of Hadley’s Hope.
Newt was the only one who escaped into the ducts.
July 3rd - Colonial Administration loses contact with Hadley's Hope colony.
July 6th - Ripley reluctantly agrees to accompany Burke and a Colonial Marine squad of 2nd Battalion, 9th Regiment to Acheron to investigate on board the USS Sulaco.
July 26th - The last pockets of resistance at Hadley's Hope are over-run by the Xenomorphs, leaving Newt as the sole survivor.
July 27th - The USS Sulaco arrives at LV-426.
(June 26 entry from the book: Alien: River of Pain, 26 June 2179.)
Edit following new Alien novels:
It is revealed in the first of the new Alien novels why it took Ripley 30+ years to get home, despite her saying in the film it will only take 6 weeks to hit the frontier. Prior to his demise, Ash uploaded a copy of his consciousness into the shuttles computer, preparing the the eventuality of his death. As soon as Ripley is asleep he re-directs the ship to drift around the cosmos searching for any signs of aliens presence, as per his orders. It is only after the events of Alien: Out of the shadows, that he is purged from the system and Ripley finally can go home.
Therefore, while technology has advanced to the point where it only takes them 3 weeks to get to LV-426, that's not as dramatic and improvement as previously thought.