widespread faucet
From the looks of your pic, the sink does have 3 holes for faucet mounting. Measure the diameter of the holes (the valve holes may be smaller than normal / and that center hole may be larger) and then go shopping for a widespread lav. faucet with valves and a spout that will fit those holes. Many of the widespread faucets on the market will fit from a 6" to 16" spread between the handles.
If the center hole is quite large, then it may be made for the old faucets with the "lift and turn" drain assebly that extened up through the spout. In that case it could be pretty expensive for a renovated faucet to fit your sink, but possibly a single lever - single hole mount faucet could be installed there. Then use hole covers on the two outside holes or use for mounting a soap/lotion dispenser.