Some of you might know how to deal with the farpoint mission, but then what would you do next? Your not going to get the next event on the very next day, so how would you know when to look for it, plus the fact yout a person out of time, so you will be handled in a different way. Who knows maybe Picard drops you off on the nearist starbase and from there yout taken back to Earth.
So really how could you keep yourself on the Enterprise?
I really would need to prove your worth to the Captain, and his past record shows he doesn't really want to know to much about his future.
We actually had a situation like that at the end of TNG season 1. Three cryogenically frozen people from close to our time are revived, stay with the crew for a time, then move on, one to descended family.
The third of those showed a bit of the way to stay on board. You're not going to get to be an ensign, not even a crewman, but you might get a bunk as a civilian consultant. Popsicle #3 was a former big business man, and understood cut-throat negotiations in a way the 24th century crew did not, making him annoying, but valuable in dealing with the Romulans. So to stay aboard, you need an asset the crew lacks or could always use more of: cut-throat negotiations, late 20th/early 21st century "historian", entertainer for 10-Forward, asst. barkeep, nursery worker (this is family oriented Galaxy-class after all), redshirt...sorry, goldshirt security muscle, etc. And if you're really willing, engineering could always use someone to scrub the ducts.
Timing would be valuable, too. The farther away from a star base when you board, the more accommodating the crew will be towards training you in some basic job that keeps you on board. Considering you are starting at Farpoint, I'm guessing that's far enough to get the ball rolling.
I'm reminded of a book where the main character has been in stasis for a hundred years or so and when he's revived, everyone's forgotten the old methods of fighting, so when he reintroduces them with his side, the other one doesn't know how to handle them... What does that have to do with anything? Since this is S1, I'd put in for Security. Help Tasha teach these 'redshirt' idiots how to fight dirty and stay alive. Sabotage the shuttles so Troi can't go to her little conference and end up getting Tasha killed.
After that, utilise a little-known in the 24th Century skill from the 21st known as 'Seat of the Pants Improvisation'.
Get people to LISTEN to Q when he's trying to get us to figure something out instead of just getting childish about his methods...
Ensure Duras has himself an 'accident' before K'mpec's death...
Lead a mutiny against Jellico... (Offer the Cardassians an exchange; Jellico for Picard...)
I know exactly what I'd do though. I'd tell the truth; that Q took me from the year 2013 and brought me to (what would be) the future. After Starfleet Security figures out there is little they can do about it, I'd jump on a Transport, head out to the Trill homewold, find Jadzia and marry her.