navigation - How would an interstellar probe navigate the pull of gravity from stars and other large objects on the way?

Others have already clarified that the chance of encountering another massive body in interstellar space is astronomically small. However, there is still something to consider for course corrections, which is that a very small error in velocity early in your transfer orbit to Alpha Centauri can result in a huge error in your intercept with the system.

Think of it like somebody firing a single atom while in orbit around Earth, trying to set up the trajectory so that hundreds of years later, it precisely hits a microscopic target that was traveling in a different orbit. For extra fun, also try to take into consideration miniscule perturbing effects that cause the orbits to deviate from their idealized model over that period of time.

In practice, the above means that you do need some way to perform course corrections if you're going to intercept your target. As you say, 'real-time' remote control is not feasible, so what this means is that the probe would need some automated control systems to gather information and autonomously correct course along the way.

There's a trade-off in when you make the corrections. The earlier you make them, the less delta-V (and hence fuel) required. But, due to the mindbogglingly huge distances involved, the more precise/accurate you need to be. In practice, you probably schedule a number of short burns along the way, so that you keep the delta-V required for your 'late' adjustments to a minimum. How you schedule those adjustments is likely a matter of careful optimization based on estimated error magnitudes.