The probes that use current technology to get to Pluto or other outer solar system objects cannot slow down enough for orbit insertion, so they can only flyby. But would Starship be able to slow down enough?
In theory New Horizons could have entered the orbit of Pluto. The delta-v required to enter orbit around Pluto is not much different from that required to escape the solar system.
The problem is it would have taken 45 years. We didn't want to wait that long, so New Horizons took a more energetic route that only took 9 years, but which put it on a path to leave the solar system.
A Hohmann transfer is a simple, efficient way to transfer between the orbit of one planet to another. There are better methods, but this is good for a simple demonstration. Feel free to go into details, but to get from low Earth orbit to Pluto orbit requires about 16km/s, just below solar escape velocity. New Horizons went 16.5 km/s, just over solar escape velocity.
The difference is New Horizons was going 16.5 km/s right from the start. A transfer orbit requires you start at a much slower 12 km/s and then slow down when you get there. It's also a longer distance because of the slower speed results in a large curve. In the diagram below, 2 is the transfer.
Whereas New Horizons was far more direct.
The other issue is New Horizons was able to expend most of its fuel right upon leaving Earth. This made the spacecraft much lighter. OTOH with a transfer orbit it would have to hold on to a portion of its fuel to brake.
A Hohmann Transfer from Earth to Pluto requires about 12 km/s to start, and then Starship would need to retain enough fuel for about 4 km/s to enter orbit. That significantly increases the mass one has to accelerate and decelerate requiring a lot more fuel. Other folks have done the math on that in a related question.
It would have to retain its fuel and relight its engines after 45 years. This is no small technical feat, and would be a high risk for such a mission.
Speaking of the rocket equation, one major advantage that New Horizons has over Starship is New Horizons used just 400 kg of spacecraft to deliver its 30 kg payload. Starship has to drag 5000 kg of Starship with it meaning significantly more fuel both to get to Pluto and to slow down.