It makes more sense to think in turn of accumulated deltaV instead of mere velocity, so you can take the potential energy from Sun's gravity into account. At the moment the fastest spacecraft ever made by Man reached 70 km/s and even that was by expending potential energy instead of building it up.
The solar escape velocity from low earth orbit is 30.9 km/s, this is how much velocity you need to turn into potential energy (altitude from the Earth then from Sun) to get out of here - the amount of velocity you need to build up right from the start, just to barely get started on the journey. So basically, your actual travel speed to your destination system will pretty much be your spacecraft's total acceleration potential halved (the other half of it being needed to decelerate once there, unless you're reasonably sure you can perform an aerocapture at the destination planet - so far this has never succeeded) minus those 30.9 km/s. You could also take into consideration the differential between the two systems but that's above my competence.
To travel those 12 light years, or roughly 10 kms, in, let's say, a millenia, you need that travel speed to be close to 3200 km/s, which is a couple orders of magnitude above our current technical ability. Even if you forgo any capacity to brake at the destination and opt for a single high-speed fly-by instead, even with the current highest-ISP engines we can build (likely a DS4G ion thruster: 21400 seconds ISP), you will need your spacecraft to start with an amount of fuel almost 10 million times its own final weight... including the ion thruster's own weight and the tankage for storign that fuel, of course. Staging can help you trim this down, but only by so much.
And even if you overcome this, your probe will most probably just end up feeding a pa'anuri somewhere en route anyway ;)