Wait, how would it be possible for Aragorn to be involved in anything going on during the Hobbit's timeline? Bilbo is supposed to be around 50 when the story starts. I started re-reading Lord of the Rings a couple weeks ago, and it says that Bilbo and Frodo share a birthday, and Frodo was turning 30 (age of maturity) at Bilbo's 111th. Then he doesn't even start HIS journey until he's 50. So even putting Aragorn at 20 years old during the events of the Hobbit, that would make him 110 years old once he meets up with Frodo... Have they ever mentioned that humans in the Tolkien universe live longer than us?
My understanding is that 60 years pass between the end of the Hobbit and the beginning of the Fellowship of the Ring (when Bilbo celebrates his 111th birthday). Now, there's also a number of years that actually pass between that and when Gandalf tells Frodo to flee the Shire (which appear to be just months apart in the film). IIRC, it's 17. So Aragorn would have been about 20 at the time of the events in the Hobbit.
This shed's a bit of light:
The son of Arathorn II and his wife Gilraen, Aragorn was born on 1 March, T.A. 2931. Through his ancestor Elendil (whom he closely resembled) Aragorn was a descendant of the first king of Númenor, Elros Tar-Minyatur; the twin brother of Elrond.
When Aragorn was two years old, his father was killed while pursuing orcs. Aragorn was afterwards fostered in Rivendell by Elrond. At the request of his mother, his lineage was kept secret, as she feared he would be killed like his father and grandfather if his true identity as Isildur's heir became known. Aragorn was renamed Estel ("hope" in Tolkien's invented language of Sindarin) to hide his existence from Sauron and his servants. He was not told about his heritage until he came of age in 2951.
Elrond revealed to Aragorn his true name and ancestry, and delivered to him the shards of Elendil's sword Narsil, and also the Ring of Barahir. He withheld the Sceptre of Annúminas from him until he "came of the right" to possess it. Aragorn met and fell in love with Arwen, Elrond's daughter (whom he mistook for Tinúviel), when she had returned from Lórien, her mother's homeland.
Aragorn thereafter assumed his role as the sixteenth Chieftain of the Dúnedain, the Rangers of the North, and went into the wild, living with the remnants of his people, whose kingdom had been destroyed through civil and regional wars centuries before.
Aragorn met Gandalf the Grey in 2956, and they became close friends. At Gandalf's request, the Rangers began to guard the Shire, inhabited by the diminutive and agrarian Hobbits. In the areas around the Shire and Bree he became known as "Strider".
From 2957 to 2980, Aragorn undertook great journeys, serving in the armies of King Thengel of Rohan (King Théoden's father) and of Steward Ecthelion II of Gondor (father of Denethor). His tasks helped to raise morale in the West and to counter the growing threat of Sauron and his allies, and he acquired experience that he would later put to use in the War of the Ring. Aragorn served his lords during that time under the name Thorongil (Eagle of the Star). With a small squadron of Gondorian ships, he led an assault on Umbar in 2980, burning many of the Corsairs' ships and personally slaying their lord during the Battle of the Havens. After the victory at Umbar, "Thorongil" left the field, to the dismay of his men, and went East.
So when he turned 20, he became Chieftain of the Dunedain. My understanding is that he is peripherally involved in fighting the forces of the Necromancer during this time period (which is more or less around when the Hobbit occurs). 5 years later, Gandalf asks him to guard the Shire. So it would not be out of the realm of the unusual to introduce this fact in the last of the three Hobbit films as part of the wrap up of the Necromancer plotline (now that they've figured out who he is, Gandalf sets Aragorn the task of keeping an eye on things in the Shire for him). This could nicely explain why Aragorn just happens to be hanging out at the Prancing Pony when Frodo shows up. It's not random chance. He's supposed to be there to keep an eye on things.
They could just as easily leave him out of it, but if they wanted to introduce a young Aragorn and show him going from young leader of a band of hill warriors to someone working for Gandalf and in the know in terms of Sauron, they certainly could.