If humanity really put their mind to it, went all out and built an Orion type spacecraft, we could probably get it there within 3000 years or so.

It's very helpful that the intended destination is to plunge into a black hole, this means we don't need to flip and slow down halfway, so we put more fuel mass fraction into the initial acceleration and we can just smack into it at relativistic velocities.

Transmitting back could be done by a much smaller relay satellite shot backwards some distance from the target, unfurling a massive mylar solar sail with a clever 'shutter' system, using the black hole's accretion disk or a close star as the light source, like a morse light on a ship.. We will work it out.. Of course nothing passes the event horizon but we might get some neat close up images and data for a short while.