I think it could be possible to launch the rocket from space, using a booster of some kind to get it out of Earth's gravity. The bigger problems, I think, are (1) navigation, given that we don't have precise measurements of how those planets are moving, and a rocket with such slow acceleration can't just turn on a dime; and (2) maintenance and repair during a decades-long journey at ludicrous speed.

The probe would have to be self-navigating to a certain extent, because we can't locate and measure the exoplanets with precision and won't be able to communicate in near-real-time to guide it in. It would have to be built for long endurance and have the capability to repair itself, too.

The final practical problem is that we wouldn't know for decades whether the probe had been successful, partially successful, or a total waste of time. It could land on an exoplanet and do wonderful science but not be able to communicate due to a broken radio. Or it could blow up in year one and we might not notice.

I suspect the first insterstellar probes instead will be inexpensive and disposable "fire and forget it" types launched with a booster rocket (maybe like the one Elon Musk is building) and no intention of decelerating or landing. They might spend centuries in flight and be designed to send pictures and data back to whoever might be listening in the future. If they're cheap enough and we have extra thrust, I could see rich people and governments launching these in hopes of a sort of future glory, knowing that each individual probe faces long odds but a thousand of them should produce at least a few winners.