Manually Drill Vias into PCB
It will be possible to drill these if you have a very rigid drill press and carbide bits. Back the PCB up with a sacrificial piece of laminate, get bright light and safety glasses and get very close to get the holes near the center. Use the highest spindle RPM your drill press is capable of (30,000 RPM is not too much) and feed slowly, especially when the bits break through. 

If your drill press is not rigid enough or the bits are the slightest bit dull, they will break off and can go anywhere, so safety glasses are not optional. 

If you use steel bits (use the best cobalt steel if you try this) they may work okay, but they'll tend to skate around on the pad and not be in the center. Replace them every 50 holes or so as they dull in the glass laminate. 

You'll still have to find a way to solder them on the top or add additional jumper wires from top to bottom. If that thing is a terminal block, for example, you'll have to add ugly jumper wires on the bottom to a number of the pads. It won't be pretty but it might help you debug. 

I doubt it's economical to get a machine shop to do this unless the boards are very, very expensive or you're willing to pay a lot of money to compress schedule. 

In the future, look at your files with a gerber viewer that includes NC drill files, make a checklist for what files have to be with a given number of layers PCB, and (IMHO) never do business with this supplier again.