You’ve got a pretty good handle on Breakthrough Starshot - their goal is indeed to ship off a tiny little craft, attached to a huge solar sail, and use high powered lasers to accelerate the craft to a speed much faster than what the power of the sun could do alone.
The spacecraft itself will have no thrusters on it - the whole point is to keep the spacecraft as light as possible so it’s easier to accelerate to relativistic speeds. The more massive your object, the more energy it takes in order to increase its speed, and getting any object up to fractions of the speed of light is difficult under any circumstances. The strategy is to make a very light, thin (but large) sail, so that a very powerful laser can bounce off of it, gradually pushing the sail along faster and faster. The actual science instruments would be relatively tiny, suspended in the very middle of the sail, weighing as little as possible, while still being able to do the science required. (We are a significant technological distance from being able to do accomplish any kind of interstellar solar sail.)
The complete lack of thrusters, plus the single direction we can push from, does mean that once the spacecraft gets up to 20% of the speed of light, it’s not slowing down again unless it crashes into something. The gravitational pull of Alpha Centauri is certainly present, but as that star is only a little bit larger than our own Sun, it doesn’t have the kind of extreme gravitational pull you’d need to slow the craft down as it goes past. Not without stopping it violently, through a collision, anyhow. At best, you’d swing past the star, get a few good images, and then sling your spacecraft right on through the solar system and out the other side - just like New Horizons did for its flyby of Pluto.
However, New Horizons had a bit more control over itself coming into its Pluto encounter than any solar sail craft would. New Horizons does have thrusters- which meant it could make adjustments to its flight en route to Pluto, and that it could course correct into a good path to encounter another object, out beyond Pluto.
Using only gravitational arguments, any solar sail spacecraft is doomed to speed up and then pretty much continue to cruise at that speed. However - it’s possible that with a very careful calculation of the spacecraft’s trajectory, you might be able to slow down the spacecraft with another method. Stars very clearly have strong, non-gravitational influences over the region surrounding them, once you get up close. It’s not just our star that has a solar wind that could be used to propel a solar sail - every star has a solar wind.
A recent paper worked out how the solar wind of the star you’re heading towards could help slow down a solar sail, perhaps enough to send it into orbit around its destination star. There are a bunch of limitations to this approach - the biggest one being that solar sail has to be able to endure quite a rapid deceleration - if the sail shreds, you’re not stopping the craft. The sail also has to be relatively gigantic - their calculations rely on something about 315 meters to a side, which is much larger than anything we’ve currently built. (The most massive one currently underway is 50 meters to a side.)
The orbit also has to be pretty precisely known - according to this paper, if you come too close to the star, you destroy the solar sail and crash into the star. However, you’d still have to sail into the solar system very, very, close to the star for the pressure from the star to slow down your spacecraft. It’s an extremely narrow range of allowable arrival positions, which the authors measure in solar radii. On an astronomical scale, this is tiny. You’re aiming to sail in to a solar system, so close to the star that you'd be about 20 times close to the Sun than Mercury, if you were arriving in our solar system. If your solar sail were a comet, this would put you in sungrazer territory.
The difficulty of managing this needle-threading exercise aside, there are still a number of technological problems - the calculation assumes that the solar sail is made of graphene, for its lightweight nature and relative strength. Graphene is not very reflective, though, which means that we'd have to get really good at coating extraordinarily thin layers of graphene with something reflective without making it brittle and prone to tearing. In principle, it's physically possible. It's just technologically improbable, for the moment.
Sign up for the mailing list for updates & news straight to your inbox!