Would like to have a pull out faucet here. Do you think I could drill a hole for a new style faucet?
Yes, you would need to locate and drill the proper sized center hole. Unfortunately, it looks like you are space-constrained by the size and placement of the cabinet stiles and back rail. I don't think you'll have enough space to locate that center hole without modifying the cabinet somehow.
The tool used to tighten and loosen the under-cabinet water fittings is called a "basin wrench", available at big box and hardware stores.
