I don't recommend the dremel drill press attachment, they have enough play in the mechanics that its not much better than by hand.
Due to space restrictions I too had to find a way to drill neatly, so I opted to make my own mini PCB drill press with auto motor activation (I used a micro switch with a long lever so when one of the bolt heads reached a certain point, the motor turns on). This is the result:
Most of it was made from scrap aluminium I got from work, but there's plenty of easy access materials like extruded engineer's T slot frame (also known as speed frame). This is a semi solid extruded box frame that has T slots running the length on each side.
Very handy stuff for making 3D printers and the like, and easy on the wallet for short lengths and small projects.
I recommend the 20mm x 20mm T frame as it's still rigid enough for the job but small enough to make the press compact.
Couple lengths of that, some simple draw slide rails, a lever, retention spring and a motor mount (I used a CNC rail guide to mount my motor with) and you have a quick and simple press for cheap that won't take up much room or cost much to make.
The best bit about using the T frame is, you can simply undo the bolts and rearrange everything; kind of like Lego (assuming you've not cut specifically angled pieces that is).
I'll post a simple guide on putting one together using the T frame if anyone is intersted, it needs basic tool handling skills only so just about anyone can make a PCB drill press with this stuff.