With a Nanodrop you are only going to measure total protein (or more specifically total aromatic amino acid concentration). So if your antibodies are pure and free from any other proteins, then you can determine their concentration using a Nanodrop. But if there is serum, or carrier protein or similar in the solution, then the background might be too high to measure the antibody concentration.
Yes, you can use a Nnodrop to measure the concentration of your monoclonal antibody (mAb). Measure at 280 nm, using the antibody's extinction coefficient (typically ~1.4 for 1 mg/mL IgG if not known exactly).
That may not be a valid assumption. Often with proteins at low concentrations they add some carrier protein, like BSA or glycogen or something to help preserve the antibody. If you purchased the mAB there should be an information or material sheet with it.